7625:
3597:
2450:
370:
1494:
1028:
1205:
3359:
4436:
2276:
2658:
217:
6612:
6261:
3092:
2944:(see below), are also used in the simpler framework of the circle. It is a common practice to allow for complex functions (or distributions) in these "real" spaces. The definition that follows does not distinguish between real or complex case.
2100:
711:
1334:
5417:
904:
5187:
1055:
6048:
540:
4536:
4006:
1784:
5643:
3193:
4912:
4757:
3592:{\displaystyle u(e^{i\theta })={\frac {a_{0}}{2}}+\sum _{k\geqslant 1}a_{k}\cos(k\theta )+b_{k}\sin(k\theta )\longrightarrow v(e^{i\theta })=\sum _{k\geqslant 1}a_{k}\sin(k\theta )-b_{k}\cos(k\theta ).}
4094:
4289:
1965:
828:
2445:{\displaystyle G(z)=c\,\exp \left({\frac {1}{2\pi }}\int _{-\pi }^{\pi }{\frac {e^{i\theta }+z}{e^{i\theta }-z}}\log \!\left(\varphi \!\left(e^{i\theta }\right)\right)\,\mathrm {d} \theta \right)}
1304:
4257:
It is possible to define Hardy spaces on other domains than the disc, and in many applications Hardy spaces on a complex half-plane (usually the right half-plane or upper half-plane) are used.
2744:
5851:
4646:
6368:
365:{\displaystyle \sup _{0\leqslant r<1}\left({\frac {1}{2\pi }}\int _{0}^{2\pi }\left|f\left(re^{i\theta }\right)\right|^{p}\;\mathrm {d} \theta \right)^{\frac {1}{p}}<\infty .}
2557:
6475:
2513:
428:
6498:
6112:
2968:
1677:
1612:
1571:
1534:
880:
743:
2533:
2481:
1489:{\displaystyle f\left(re^{i\theta }\right)={\frac {1}{2\pi }}\int _{0}^{2\pi }P_{r}(\theta -\phi ){\tilde {f}}\left(e^{i\phi }\right)\,\mathrm {d} \phi ,\quad r<1,}
1988:
1023:{\displaystyle g\in H^{p}\left(\mathbf {T} \right){\text{ if and only if }}g\in L^{p}\left(\mathbf {T} \right){\text{ and }}{\hat {g}}(n)=0{\text{ for all }}n<0,}
618:
5315:
1876:< â, the real Hardy space contains the Hardy space, but is much bigger, since no relationship is imposed between the real and imaginary part of the function.
2141:) (defined below) in the simple way given above, but must use the actual definition using maximal functions, which is given further along somewhere below.
4556:, they are not interchangeable as domains for Hardy spaces. Contributing to this difference is the fact that the unit circle has finite (one-dimensional)
1200:{\displaystyle \forall n\in \mathbf {Z} ,\ \ \ {\hat {g}}(n)={\frac {1}{2\pi }}\int _{0}^{2\pi }g\left(e^{i\phi }\right)e^{-in\phi }\,\mathrm {d} \phi .}
7514:
5051:
6736:
4793:
6698:
7350:
6929:
7177:
5985:
451:
4455:
3912:
1883:< 1, such tools as Fourier coefficients, Poisson integral, conjugate function, are no longer valid. For example, consider the function
1685:
5554:
7340:
6289:
Martingales with continuous time parameter can also be considered. A direct link with the classical theory is obtained via the complex
3137:
4812:
3899:
7027:
5924:
7467:
7322:
4677:
7298:
4431:{\displaystyle \|f\|_{H^{p}}=\sup _{y>0}\left(\int _{-\infty }^{+\infty }|f(x+iy)|^{p}\,\mathrm {d} x\right)^{\frac {1}{p}}.}
4032:
7129:
7063:
6995:
6970:
6816:
6794:
1889:
755:
6730:
Burkholder, Donald L.; Gundy, Richard F.; Silverstein, Martin L. (1971), "A maximal function characterization of the class
1256:
2684:
7190:
589:
The Hardy spaces defined in the preceding section can also be viewed as certain closed vector subspaces of the complex
7279:
7170:
7037:
5786:
4584:
7549:
7664:
7194:
6060:
5743:
2807:
be an outer function represented as above from a function Ï on the circle. Replacing Ï by Ï, α > 0, a family (
2255:
factorization" allows the Hardy space to be completely characterized by the spaces of inner and outer functions.
6318:
85:
on the real line, which are (in the sense of distributions) boundary values of the holomorphic functions of the
1789:
In applications, those functions with vanishing negative
Fourier coefficients are commonly interpreted as the
7345:
7149:
6895:
6786:
5943:
4994:(this depends on the choice of Ί, but different choices of Schwartz functions Ί give equivalent norms). The
2653:{\displaystyle \lim _{r\to 1^{-}}\left|G\left(re^{i\theta }\right)\right|=\varphi \left(e^{i\theta }\right)}
7628:
7401:
7335:
7163:
6808:
7365:
7144:
6890:
7054:, Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series), Basel:
6431:
7610:
7564:
7488:
7370:
7016:
2178:
82:
6607:{\displaystyle \int _{0}^{1}{\Bigl (}\sum |c_{k}h_{k}(x)|^{2}{\Bigr )}^{\frac {1}{2}}\,\mathrm {d} x.}
6256:{\displaystyle S(f)=\left(|M_{0}|^{2}+\sum _{n=0}^{\infty }|M_{n+1}-M_{n}|^{2}\right)^{\frac {1}{2}}.}
3087:{\displaystyle (Mf)(e^{i\theta })=\sup _{0<r<1}\left|(f*P_{r})\left(e^{i\theta }\right)\right|,}
7654:
7605:
7421:
7085:
2486:
1840:
discussed further down in this article are easy to describe in the present context. A real function
566:
390:
7659:
7457:
7355:
7258:
7139:
6406:
is the finite field generated by the dyadic partition of into 2 intervals of length 2, for every
5221:
6885:
4553:
1653:
1588:
1547:
1510:
856:
719:
127:
There are also higher-dimensional generalizations, consisting of certain holomorphic functions on
7554:
7330:
7585:
7529:
7493:
2095:{\displaystyle f(e^{i\theta }):=\lim _{r\to 1}F(re^{i\theta })=i\,\cot({\tfrac {\theta }{2}}).}
706:{\displaystyle {\tilde {f}}\left(e^{i\theta }\right)=\lim _{r\to 1}f\left(re^{i\theta }\right)}
5247:
has elements that are not functions, and its dual is the homogeneous
Lipschitz space of order
2518:
2466:
7649:
7292:
5412:{\displaystyle \int _{\mathbf {R} ^{n}}f(x)x_{1}^{i_{1}}\ldots x_{n}^{i_{n}}\,\mathrm {d} x,}
578:
139:
20:
7288:
5693: < 1 (on the circle, the corresponding representation is valid for 0 <
4156: < 1. Distributions that are not functions do occur, as is seen with functions
7568:
7073:
6914:
6876:
6765:
1814:
1042:
445:
is defined as the vector space of bounded holomorphic functions on the disk, with the norm
166:
47:
7155:
6950:
379:
is a vector space. The number on the left side of the above inequality is the Hardy space
8:
7534:
7472:
7186:
97:
43:
3795:= â was excluded from the definition of real Hardy spaces, because the maximal function
7559:
7426:
7102:
6987:
6979:
6962:
6769:
6753:
2184:
124:
spaces have some undesirable properties, and the Hardy spaces are much better behaved.
6712:
6693:
6308:= 0. Let Ï denote the hitting time of the unit circle. For every holomorphic function
7539:
7125:
7106:
7059:
7033:
6991:
6966:
6840:
6812:
6790:
4800:
3627:
1802:
1311:
574:
154:
6773:
7544:
7462:
7431:
7411:
7396:
7391:
7386:
7094:
6946:
6938:
6862:
6853:
6745:
6707:
6689:
6668:
6659:
5763:. Assume for simplicity that ÎŁ is equal to the Ï-field generated by the sequence (ÎŁ
4804:
4557:
4269:
2766:
185:
55:
27:
7223:
3811:. However, the two following properties are equivalent for a real valued function
7406:
7360:
7308:
7303:
7274:
7069:
7023:
6911:
6872:
6761:
1872:) belong to the space (see the section on real Hardy spaces below). Thus for 1 â€
7233:
5182:{\displaystyle f_{k}(x)=\mathbf {1} _{}(x-k)-\mathbf {1} _{}(x+k),\ \ \ k>0.}
2796:), where Ï is the positive function in the representation of the outer function
7595:
7447:
7248:
6942:
6907:
6833:
6630:
6290:
2754:
2544:
1319:
143:
86:
4152:
Distributions on the circle are general enough for handling Hardy spaces when
7643:
7600:
7524:
7253:
7238:
7228:
7080:
2252:
75:
59:
7055:
7590:
7243:
7213:
7113:
6415:
5682:
3643:
1235:
2206:
2191:= 1. The Dirac distribution at a point of the unit circle belongs to real-
596:
on the unit circle. This connection is provided by the following theorem (
7519:
7416:
7218:
7121:
6844:
6825:
4766:
128:
67:
7452:
7284:
7098:
6920:
6867:
6757:
6673:
6654:
6043:{\displaystyle M_{n}=\operatorname {E} {\bigl (}f|\Sigma _{n}{\bigr )}}
2750:
147:
3803:
function is always bounded, and because it is not desirable that real-
535:{\displaystyle \|f\|_{H^{\infty }}=\sup _{|z|<1}\left|f(z)\right|.}
4959:
4542:
2765:. The inner function can be further factored into a form involving a
170:
51:
6924:
6749:
6655:"On two problems concerning linear transformations in Hilbert space"
4531:{\displaystyle \|f\|_{H^{\infty }}=\sup _{z\in \mathbf {H} }|f(z)|.}
4001:{\displaystyle F(z)=\sum _{n=0}^{+\infty }c_{n}z^{n},\quad |z|<1}
1779:{\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n},\ \ \ |z|<1.}
5638:{\displaystyle f=\sum c_{j}a_{j},\ \ \ \sum |c_{j}|^{p}<\infty }
4763:
4561:
590:
174:
90:
6687:
5705:†1/2 because their maximal function is equivalent at infinity to
4449:) is defined as functions of bounded norm, with the norm given by
3878:
cannot be reconstructed from the real part of its boundary limit
3188:{\displaystyle e^{i\varphi }\rightarrow P_{r}(\theta -\varphi ).}
2920:< 1, can be expressed as product of several functions in some
2121:) is 0 almost everywhere, so it is no longer possible to recover
4907:{\displaystyle (M_{\Phi }f)(x)=\sup _{t>0}|(f*\Phi _{t})(x)|}
1824:
5697: < 1, but on the line, Haar functions do not belong to
5669:
On the line for example, the difference of Dirac distributions
1790:
550:
109:
2931:
2129:). As a consequence of this example, one sees that for 0 <
5281: < 1 and so defines a metric on the Hardy space
4752:{\displaystyle (Mf)(z):={\frac {\sqrt {\pi }}{1-z}}f(m(z)).}
2936:
Real-variable techniques, mainly associated to the study of
131:
in the complex case, or certain spaces of distributions on
7185:
7083:(1923), "Ăber die Randwerte einer analytischen Funktion",
6925:"On the mean value of the modulus of an analytic function"
6729:
6386:
6094: > 1) and the Burgess Davis inequality (when
4099:
converges in the sense of distributions to a distribution
6904:
5263:-quasinorm is not a norm, as it is not subadditive. The
5750:), with respect to an increasing sequence of Ï-fields (ÎŁ
4252:
4199:) iff it is the boundary value of the real part of some
4089:{\displaystyle \sum _{n=0}^{+\infty }c_{n}e^{in\theta }}
3857:
2814:) of outer functions is obtained, with the properties:
2207:
Factorization into inner and outer functions (Beurling)
1960:{\displaystyle F(z)={\frac {1+z}{1-z}},\quad |z|<1.}
823:{\displaystyle \|{\tilde {f}}\|_{L^{p}}=\|f\|_{H^{p}}.}
5444: â 1), vanish. For example, the integral of
2075:
6501:
6434:
6321:
6115:
5988:
5789:
5557:
5318:
5054:
4815:
4680:
4587:
4458:
4292:
4035:
3915:
3362:
3140:
2971:
2687:
2560:
2521:
2489:
2469:
2279:
1991:
1892:
1688:
1656:
1591:
1550:
1513:
1337:
1259:
1058:
907:
859:
758:
722:
621:
454:
393:
220:
5919:; hence it converges almost surely to some function
4275:
is defined to be the space of holomorphic functions
4145:
can be computed from the
Fourier coefficients of Re(
3353:
extends to a holomorphic function in the unit disk,
19:"Hardy class" redirects here. For the warships, see
4130:
can be reconstructed from the real distribution Re(
3882:on the circle, because of the lack of convexity of
1844:on the unit circle belongs to the real Hardy space
1299:{\displaystyle {\tilde {f}}\in L^{p}(\mathbf {T} )}
7515:Spectral theory of ordinary differential equations
6606:
6469:
6362:
6255:
6042:
5845:
5637:
5411:
5181:
4906:
4751:
4640:
4530:
4430:
4191:A real distribution on the circle belongs to real-
4088:
4000:
3591:
3187:
3086:
2739:{\displaystyle \lim _{r\to 1^{-}}h(re^{i\theta })}
2738:
2652:
2527:
2515:is integrable on the circle. In particular, when
2507:
2475:
2444:
2094:
1959:
1778:
1671:
1606:
1565:
1528:
1488:
1298:
1253:The above can be turned around. Given a function
1199:
1022:
874:
822:
737:
705:
584:
534:
422:
364:
6737:Transactions of the American Mathematical Society
6576:
6519:
5521:-atom is bounded by a constant depending only on
5228:contains unbounded functions (proving again that
4134:) on the circle, because the Taylor coefficients
3658: < â, the following are equivalent for a
3626: < â (up to a scalar multiple, it is the
2896:can be expressed as the product of a function in
2401:
2392:
160:
7641:
6699:Proceedings of the American Mathematical Society
6392:
6300:) in the complex plane, starting from the point
5804:
4848:
4486:
4320:
3007:
2689:
2562:
2018:
659:
482:
222:
6694:"Inner and outer functions on Riemann surfaces"
5953:can be expressed as conditional expectation of
5846:{\displaystyle M^{*}=\sup _{n\geq 0}\,|M_{n}|.}
4641:{\displaystyle m(z)=i\cdot {\frac {1+z}{1-z}}.}
3101:indicates convolution between the distribution
2678:| †1 on the unit disk and the limit
545:For 0 < p †q †â, the class
6930:Proceedings of the London Mathematical Society
6839:
6805:Blaschke Products - Bounded Analytic Functions
6652:
3886:in this case. Convexity fails but a kind of "
138:Hardy spaces have a number of applications in
7171:
6102:-norm of the maximal function to that of the
6079:< â. The interesting space is martingale-
6035:
6010:
5963:. It is thus possible to identify martingale-
2954:denote the Poisson kernel on the unit circle
2463:| = 1, and some positive measurable function
2183:on the circle, is a non-zero multiple of the
1825:Connection to real Hardy spaces on the circle
207:< â is the class of holomorphic functions
6373:is a martingale, that belongs to martingale-
4560:while the real line does not. However, for
4466:
4459:
4300:
4293:
3338:on the unit circle, one associates the real
801:
794:
775:
759:
749:space for the unit circle, and one has that
462:
455:
401:
394:
6414:on is represented by its expansion on the
5746:on some probability space (Ω, Σ,
4772:
4283:with bounded norm, the norm being given by
2932:Real-variable techniques on the unit circle
7178:
7164:
6978:
6781:Cima, Joseph A.; Ross, William T. (2000),
6363:{\displaystyle M_{t}=F(B_{t\wedge \tau })}
3822: is the real part of some function
1852:) if it is the real part of a function in
895:
597:
328:
104: < â these real Hardy spaces
7137:
7022:
6866:
6711:
6672:
6621:(ÎŽ), is isomorphic to the classical real
6592:
5819:
5397:
5305:of compact support is in the Hardy space
4552:can be mapped to one another by means of
4400:
2428:
2298:
2133:< 1, one cannot characterize the real-
2064:
1461:
1185:
7468:Group algebra of a locally compact group
6780:
6387:Burkholder, Gundy & Silverstein 1971
6090:The BurkholderâGundy inequalities (when
5544:as a convergent infinite combination of
3131:-function defined on the unit circle by
2543:because the above takes the form of the
2215: †â, every non-zero function
716:exists for almost every Ξ. The function
7029:Representation Theorems in Hardy Spaces
6956:
6883:
6802:
5292:
5285:, which defines the topology and makes
5243: < 1 then the Hardy space
3334:To every real trigonometric polynomial
3283: ≥ 1, the distribution
1864:belongs to the real Hardy space iff Re(
7642:
7045:
7003:
6901:
6626:
6283:
6084:
4026:). It follows that the Fourier series
7159:
7112:
7079:
6919:
6824:
6783:The Backward Shift on the Hardy Space
4780:In analysis on the real vector space
4253:Hardy spaces for the upper half plane
3610:extends to a bounded linear operator
3329:
2548:
2248:
89:Hardy spaces, and are related to the
71:
63:
6984:An Introduction to Harmonic Analysis
6906:, Mathematics Lecture Notes Series,
4566:, one has the following theorem: if
4217:of the unit circle, belongs to real-
3123:)(e) is the result of the action of
853:) consisting of all limit functions
6959:Banach Spaces of Analytic Functions
5301:†1, a bounded measurable function
2908:is the product of two functions in
2757:is equal to 1 a.e. In particular,
13:
6594:
6470:{\displaystyle f=\sum c_{k}h_{k},}
6183:
6024:
6002:
5681:can be represented as a series of
5632:
5505:| denotes the Euclidean volume of
5399:
4878:
4824:
4578:denotes the Möbius transformation
4475:
4402:
4357:
4349:
4055:
3950:
3674:is the real part of some function
2860:| almost everywhere on the circle.
2430:
1720:
1463:
1187:
1059:
471:
356:
330:
14:
7676:
6713:10.1090/S0002-9939-1965-0183883-1
6617:This space, sometimes denoted by
2904:. For example: every function in
2551:, Thm 17.16). This implies that
1318:on the unit disk by means of the
211:on the open unit disk satisfying
7624:
7623:
7550:Topological quantum field theory
6083:, whose dual is martingale-BMO (
5780:of the martingale is defined by
5720:
5525:and on the Schwartz function Ί.
5326:
5121:
5079:
4497:
4239:< 1/2, second derivatives ÎŽâČâČ
3890:" remains, namely the fact that
3299:), namely the boundary value of
2864:It follows that whenever 0 <
1797:is seen to sit naturally inside
1289:
1069:
963:
929:
199:More generally, the Hardy space
5309:if and only if all its moments
3978:
2203: < 1 (see below).
1937:
1473:
1049:integrable on the unit circle,
585:Hardy spaces on the unit circle
161:Hardy spaces for the unit disk
16:Concept within complex analysis
7032:, Cambridge University Press,
6851:spaces of several variables",
6681:
6646:
6563:
6558:
6552:
6528:
6357:
6338:
6270:can be defined by saying that
6224:
6189:
6154:
6138:
6125:
6119:
6019:
5925:martingale convergence theorem
5836:
5821:
5689:-quasinorm when 1/2 <
5619:
5603:
5347:
5341:
5289:into a complete metric space.
5155:
5143:
5138:
5126:
5113:
5101:
5096:
5084:
5071:
5065:
5013: < â, the Hardy space
4925:), where â is convolution and
4900:
4896:
4890:
4887:
4868:
4864:
4841:
4835:
4832:
4816:
4743:
4740:
4734:
4728:
4699:
4693:
4690:
4681:
4597:
4591:
4521:
4517:
4511:
4504:
4390:
4385:
4370:
4363:
3988:
3980:
3925:
3919:
3752:), hence the real Hardy space
3583:
3574:
3552:
3543:
3505:
3489:
3483:
3480:
3471:
3449:
3440:
3382:
3366:
3179:
3167:
3154:
3052:
3033:
3000:
2984:
2981:
2972:
2733:
2714:
2696:
2569:
2508:{\displaystyle \log(\varphi )}
2502:
2496:
2289:
2283:
2223:can be written as the product
2086:
2071:
2055:
2036:
2025:
2011:
1995:
1947:
1939:
1902:
1896:
1766:
1758:
1698:
1692:
1663:
1598:
1557:
1520:
1434:
1425:
1413:
1293:
1285:
1266:
1103:
1097:
1091:
994:
988:
982:
866:
768:
729:
666:
628:
521:
515:
495:
487:
434:â„ 1, but not when 0 <
423:{\displaystyle \|f\|_{H^{p}}.}
1:
7346:Uniform boundedness principle
6787:American Mathematical Society
6723:
5944:dominated convergence theorem
5033:. One can find sequences in
5021:, with equivalent norm. When
4803:Ί with â«ÎŠ = 1, the
3906:> 0. As a consequence, if
3858:Real Hardy spaces for 0 <
3112:(Ξ) on the circle. Namely, (
2535:is integrable on the circle,
2483:on the unit circle such that
1982:< 1, and the radial limit
1801:space, and is represented by
6809:University of Michigan Press
6492:norm of the square function
5200:norms are not equivalent on
5017:is the same vector space as
4792: †â) consists of
3708:the radial maximal function
3236:defined on the unit disk by
3213:) consists of distributions
2788:if and only if Ï belongs to
1793:solutions. Thus, the space
1679:is the holomorphic function
1672:{\displaystyle {\tilde {f}}}
1607:{\displaystyle {\tilde {f}}}
1566:{\displaystyle {\tilde {f}}}
1529:{\displaystyle {\tilde {f}}}
875:{\displaystyle {\tilde {f}}}
833:Denoting the unit circle by
738:{\displaystyle {\tilde {f}}}
7:
7145:Encyclopedia of Mathematics
7046:MĂŒller, Paul F. X. (2005),
6902:Garsia, Adriano M. (1973),
6891:Encyclopedia of Mathematics
6400:In this example, Ω = and Σ
6393:Example: dyadic martingale-
5517:-quasinorm of an arbitrary
5025: = 1, the Hardy space
3772:= 1, the real Hardy space
10:
7681:
7489:Invariant subspace problem
7017:Cambridge University Press
6629:). The Haar system is an
5448:must vanish in order that
5045:, for example on the line
4998:-quasinorm is a norm when
4172:| < 1), that belong to
3841: and its conjugate
3780:) is a proper subspace of
1860:), and a complex function
1614:has Fourier coefficients (
1218:) is a closed subspace of
939: if and only if
180:consists of the functions
58:. They were introduced by
18:
7619:
7578:
7502:
7481:
7440:
7379:
7321:
7267:
7209:
7202:
7138:Shvedenko, S.V. (2001) ,
7118:Real and Complex Analysis
7086:Mathematische Zeitschrift
6957:Hoffman, Kenneth (1988),
6061:Doob's maximal inequality
5979:such that the martingale
5482:has support in some ball
5475:this is also sufficient.
4651:Then the linear operator
4548:and the upper half-plane
3630:on the unit circle), and
2672:inner (interior) function
2268:outer (exterior) function
1642:< 0, then the element
845:) the vector subspace of
561:-norm is increasing with
7458:Spectrum of a C*-algebra
6943:10.1112/plms/s2_14.1.269
6640:
6063:implies that martingale-
5860:< â. The martingale (
5222:bounded mean oscillation
5029:is a proper subspace of
4207:. A Dirac distribution ÎŽ
4103:on the unit circle, and
2962:on the unit circle, set
2528:{\displaystyle \varphi }
2476:{\displaystyle \varphi }
2455:for some complex number
890:, one then has that for
573:-norm is increasing for
565:(it is a consequence of
188:on the circle of radius
70:, because of the paper (
66:), who named them after
7555:Noncommutative geometry
6884:Folland, G.B. (2001) ,
6803:Colwell, Peter (1985),
6653:Beurling, Arne (1948).
4015:, it can be shown that
3261:radial maximal function
1310:â„ 1, one can regain a (
7665:Schwartz distributions
7611:TomitaâTakesaki theory
7586:Approximation property
7530:Calculus of variations
6608:
6488:can be defined by the
6471:
6364:
6257:
6187:
6053:belongs to martingale-
6044:
5975:) consisting of those
5847:
5639:
5413:
5255: â 1). When
5183:
4908:
4794:tempered distributions
4788:(for 0 <
4773:Real Hardy spaces for
4753:
4642:
4554:Möbius transformations
4532:
4432:
4229:< 1; derivatives ÎŽâČ
4188:an integer â„ 1).
4090:
4059:
4002:
3954:
3654:. When 1 ≤
3593:
3255:)(e) is harmonic, and
3189:
3088:
2740:
2654:
2529:
2509:
2477:
2446:
2144:For the same function
2096:
1961:
1780:
1724:
1673:
1608:
1567:
1530:
1490:
1300:
1201:
1024:
876:
824:
739:
707:
612:â„ 1, the radial limit
536:
424:
366:
142:itself, as well as in
81:are certain spaces of
7606:BanachâMazur distance
7569:Generalized functions
7048:Isomorphisms Between
6625:space on the circle (
6609:
6472:
6365:
6258:
6167:
6045:
5967:with the subspace of
5848:
5640:
5414:
5184:
4980:is defined to be the
4909:
4754:
4643:
4533:
4433:
4249:< 1/3, and so on.
4091:
4036:
4003:
3931:
3594:
3307: ≥ 1, the
3190:
3105:and the function e â
3089:
2958:. For a distribution
2741:
2655:
2530:
2510:
2478:
2447:
2270:if it takes the form
2211:For 0 <
2097:
1962:
1815:bi-infinite sequences
1781:
1704:
1674:
1609:
1568:
1531:
1491:
1301:
1202:
1025:
877:
825:
740:
708:
537:
425:
367:
167:holomorphic functions
140:mathematical analysis
48:holomorphic functions
21:Hardy class destroyer
7351:Kakutani fixed-point
7336:Riesz representation
6499:
6480:then the martingale-
6432:
6319:
6113:
5986:
5787:
5555:
5542:atomic decomposition
5460:†1, and as long as
5316:
5293:Atomic decomposition
5052:
5037:that are bounded in
4813:
4678:
4585:
4456:
4290:
4122:)(Ξ). The function
4033:
3913:
3768:) in this case. For
3712: belongs to
3666:on the unit circle:
3662:integrable function
3622:), when 1 <
3360:
3340:conjugate polynomial
3138:
2969:
2912:; every function in
2685:
2663:for almost every Ξ.
2558:
2519:
2487:
2467:
2277:
2251:, Thm 17.17). This "
2247:, as defined below (
1989:
1890:
1686:
1654:
1589:
1548:
1511:
1335:
1257:
1056:
1043:Fourier coefficients
905:
857:
756:
720:
619:
581:with total mass 1).
575:probability measures
452:
391:
218:
100:. For 1 â€
7535:Functional calculus
7494:Mahler's conjecture
7473:Von Neumann algebra
7187:Functional analysis
7015:(Second ed.),
7004:Koosis, P. (1998),
6980:Katznelson, Yitzhak
6688:Voichick, Michael;
6631:unconditional basis
6516:
6410:â„ 0. If a function
5486:and is bounded by |
5396:
5371:
5277:is subadditive for
5006: < 1.
4799:such that for some
4769:of Hilbert spaces.
4361:
3870:< 1, a function
3271: belongs to
2928: > 1.
2342:
1646:of the Hardy space
1402:
1141:
1005: for all
600:, Thm 3.8): Given
567:Hölder's inequality
438: < 1.
281:
192:remains bounded as
98:functional analysis
7560:Riemann hypothesis
7259:Topological vector
7099:10.1007/BF01192397
6988:Dover Publications
6963:Dover Publications
6868:10.1007/BF02392215
6841:Fefferman, Charles
6830:Theory of H-Spaces
6674:10.1007/BF02395019
6604:
6502:
6467:
6360:
6312:in the unit disk,
6253:
6106:of the martingale
6040:
5901:, the martingale (
5843:
5818:
5635:
5409:
5375:
5350:
5179:
5002:â„ 1, but not when
4972:of a distribution
4904:
4862:
4784:, the Hardy space
4749:
4638:
4528:
4502:
4441:The corresponding
4428:
4341:
4334:
4184: < 1 (and
4086:
3998:
3693:and its conjugate
3589:
3526:
3423:
3330:Conjugate function
3202: < â, the
3185:
3084:
3027:
2900:and a function in
2736:
2710:
2650:
2583:
2525:
2505:
2473:
2442:
2325:
2185:Dirac distribution
2163:). The limit when
2092:
2084:
2032:
1957:
1803:infinite sequences
1776:
1669:
1604:
1563:
1544:). Supposing that
1526:
1486:
1385:
1296:
1197:
1124:
1020:
872:
820:
735:
703:
673:
532:
506:
430:It is a norm when
420:
362:
264:
242:
135:in the real case.
7637:
7636:
7540:Integral operator
7317:
7316:
7131:978-0-07-100276-9
7065:978-3-7643-2431-5
6997:978-0-486-63331-2
6972:978-0-486-65785-1
6818:978-0-472-10065-1
6796:978-0-8218-2083-4
6690:Zalcman, Lawrence
6589:
6247:
6071:(Ω, Σ,
5971:(Ω, Σ,
5803:
5717: â 0).
5598:
5595:
5592:
5532:†1, any element
5232:is not closed in
5208:is not closed in
5169:
5166:
5163:
5041:but unbounded in
4847:
4801:Schwartz function
4723:
4711:
4633:
4485:
4422:
4319:
4176:when 0 <
3888:complex convexity
3760:) coincides with
3628:Hilbert transform
3511:
3408:
3403:
3318:) is a subset of
3006:
2938:real Hardy spaces
2888:, every function
2688:
2561:
2387:
2323:
2083:
2017:
1978:for every 0 <
1932:
1835:real Hardy spaces
1756:
1753:
1750:
1666:
1601:
1560:
1523:
1437:
1383:
1269:
1122:
1094:
1084:
1081:
1078:
1006:
985:
974:
940:
869:
771:
732:
658:
631:
481:
350:
262:
221:
186:mean square value
155:scattering theory
7672:
7655:Complex analysis
7627:
7626:
7545:Jones polynomial
7463:Operator algebra
7207:
7206:
7180:
7173:
7166:
7157:
7156:
7152:
7134:
7109:
7076:
7042:
7019:
7006:Introduction to
7000:
6975:
6953:
6910:
6898:
6880:
6870:
6861:(3â4): 137â193,
6854:Acta Mathematica
6836:
6821:
6799:
6777:
6718:
6717:
6715:
6706:(6): 1200â1204.
6685:
6679:
6678:
6676:
6660:Acta Mathematica
6650:
6613:
6611:
6610:
6605:
6597:
6591:
6590:
6582:
6580:
6579:
6572:
6571:
6566:
6551:
6550:
6541:
6540:
6531:
6523:
6522:
6515:
6510:
6476:
6474:
6473:
6468:
6463:
6462:
6453:
6452:
6369:
6367:
6366:
6361:
6356:
6355:
6331:
6330:
6262:
6260:
6259:
6254:
6249:
6248:
6240:
6238:
6234:
6233:
6232:
6227:
6221:
6220:
6208:
6207:
6192:
6186:
6181:
6163:
6162:
6157:
6151:
6150:
6141:
6098:= 1) relate the
6049:
6047:
6046:
6041:
6039:
6038:
6032:
6031:
6022:
6014:
6013:
5998:
5997:
5852:
5850:
5849:
5844:
5839:
5834:
5833:
5824:
5817:
5799:
5798:
5778:maximal function
5685:, convergent in
5644:
5642:
5641:
5636:
5628:
5627:
5622:
5616:
5615:
5606:
5596:
5593:
5590:
5586:
5585:
5576:
5575:
5474:
5469: / (
5418:
5416:
5415:
5410:
5402:
5395:
5394:
5393:
5383:
5370:
5369:
5368:
5358:
5337:
5336:
5335:
5334:
5329:
5220:of functions of
5188:
5186:
5185:
5180:
5167:
5164:
5161:
5142:
5141:
5124:
5100:
5099:
5082:
5064:
5063:
4953:
4939:
4913:
4911:
4910:
4905:
4903:
4886:
4885:
4867:
4861:
4828:
4827:
4805:maximal function
4758:
4756:
4755:
4750:
4724:
4722:
4707:
4706:
4647:
4645:
4644:
4639:
4634:
4632:
4621:
4610:
4558:Lebesgue measure
4537:
4535:
4534:
4529:
4524:
4507:
4501:
4500:
4481:
4480:
4479:
4478:
4437:
4435:
4434:
4429:
4424:
4423:
4415:
4413:
4409:
4405:
4399:
4398:
4393:
4366:
4360:
4352:
4333:
4315:
4314:
4313:
4312:
4270:upper half-plane
4260:The Hardy space
4095:
4093:
4092:
4087:
4085:
4084:
4069:
4068:
4058:
4050:
4007:
4005:
4004:
3999:
3991:
3983:
3974:
3973:
3964:
3963:
3953:
3945:
3598:
3596:
3595:
3590:
3567:
3566:
3536:
3535:
3525:
3504:
3503:
3464:
3463:
3433:
3432:
3422:
3404:
3399:
3398:
3389:
3381:
3380:
3309:real Hardy space
3291:" a function in
3204:real Hardy space
3198:For 0 <
3194:
3192:
3191:
3186:
3166:
3165:
3153:
3152:
3093:
3091:
3090:
3085:
3080:
3076:
3075:
3071:
3070:
3051:
3050:
3026:
2999:
2998:
2776:, decomposed as
2767:Blaschke product
2745:
2743:
2742:
2737:
2732:
2731:
2709:
2708:
2707:
2674:if and only if |
2659:
2657:
2656:
2651:
2649:
2645:
2644:
2622:
2618:
2617:
2613:
2612:
2611:
2582:
2581:
2580:
2534:
2532:
2531:
2526:
2514:
2512:
2511:
2506:
2482:
2480:
2479:
2474:
2451:
2449:
2448:
2443:
2441:
2437:
2433:
2427:
2423:
2422:
2418:
2417:
2388:
2386:
2379:
2378:
2365:
2358:
2357:
2344:
2341:
2336:
2324:
2322:
2311:
2105:exists for a.e.
2101:
2099:
2098:
2093:
2085:
2076:
2054:
2053:
2031:
2010:
2009:
1966:
1964:
1963:
1958:
1950:
1942:
1933:
1931:
1920:
1909:
1785:
1783:
1782:
1777:
1769:
1761:
1754:
1751:
1748:
1744:
1743:
1734:
1733:
1723:
1718:
1678:
1676:
1675:
1670:
1668:
1667:
1659:
1613:
1611:
1610:
1605:
1603:
1602:
1594:
1572:
1570:
1569:
1564:
1562:
1561:
1553:
1535:
1533:
1532:
1527:
1525:
1524:
1516:
1495:
1493:
1492:
1487:
1466:
1460:
1456:
1455:
1439:
1438:
1430:
1412:
1411:
1401:
1393:
1384:
1382:
1371:
1366:
1362:
1361:
1360:
1305:
1303:
1302:
1297:
1292:
1284:
1283:
1271:
1270:
1262:
1206:
1204:
1203:
1198:
1190:
1184:
1183:
1165:
1161:
1160:
1140:
1132:
1123:
1121:
1110:
1096:
1095:
1087:
1082:
1079:
1076:
1072:
1029:
1027:
1026:
1021:
1007:
1004:
987:
986:
978:
975:
972:
970:
966:
957:
956:
941:
938:
936:
932:
923:
922:
894: â„ 1,(
881:
879:
878:
873:
871:
870:
862:
829:
827:
826:
821:
816:
815:
814:
813:
790:
789:
788:
787:
773:
772:
764:
744:
742:
741:
736:
734:
733:
725:
712:
710:
709:
704:
702:
698:
697:
696:
672:
654:
650:
649:
633:
632:
624:
541:
539:
538:
533:
528:
524:
505:
498:
490:
477:
476:
475:
474:
429:
427:
426:
421:
416:
415:
414:
413:
371:
369:
368:
363:
352:
351:
343:
341:
337:
333:
327:
326:
321:
317:
316:
312:
311:
310:
280:
272:
263:
261:
250:
241:
196:â 1 from below.
56:upper half plane
28:complex analysis
7680:
7679:
7675:
7674:
7673:
7671:
7670:
7669:
7660:Operator theory
7640:
7639:
7638:
7633:
7615:
7579:Advanced topics
7574:
7498:
7477:
7436:
7402:HilbertâSchmidt
7375:
7366:GelfandâNaimark
7313:
7263:
7198:
7184:
7140:"Hardy classes"
7132:
7066:
7040:
7011:
6998:
6973:
6845:Stein, Elias M.
6819:
6797:
6750:10.2307/1995838
6726:
6721:
6686:
6682:
6651:
6647:
6643:
6593:
6581:
6575:
6574:
6573:
6567:
6562:
6561:
6546:
6542:
6536:
6532:
6527:
6518:
6517:
6511:
6506:
6500:
6497:
6496:
6458:
6454:
6448:
6444:
6433:
6430:
6429:
6423:
6405:
6398:
6345:
6341:
6326:
6322:
6320:
6317:
6316:
6298:
6291:Brownian motion
6239:
6228:
6223:
6222:
6216:
6212:
6197:
6193:
6188:
6182:
6171:
6158:
6153:
6152:
6146:
6142:
6137:
6136:
6132:
6131:
6114:
6111:
6110:
6104:square function
6067:coincides with
6034:
6033:
6027:
6023:
6018:
6009:
6008:
5993:
5989:
5987:
5984:
5983:
5962:
5951:
5932:
5914:
5906:
5873:
5865:
5835:
5829:
5825:
5820:
5807:
5794:
5790:
5788:
5785:
5784:
5775:
5768:
5762:
5755:
5741:
5733:
5726:
5680:
5676:
5664:
5659:-atoms and the
5653:
5623:
5618:
5617:
5611:
5607:
5602:
5581:
5577:
5571:
5567:
5556:
5553:
5552:
5478:If in addition
5465:
5434:
5428:
5398:
5389:
5385:
5384:
5379:
5364:
5360:
5359:
5354:
5330:
5325:
5324:
5323:
5319:
5317:
5314:
5313:
5295:
5276:
5212:. The dual of
5125:
5120:
5119:
5083:
5078:
5077:
5059:
5055:
5053:
5050:
5049:
5009:If 1 <
4990:
4971:
4948: /
4940:
4933:
4926:
4899:
4881:
4877:
4863:
4851:
4823:
4819:
4814:
4811:
4810:
4778:
4712:
4705:
4679:
4676:
4675:
4622:
4611:
4609:
4586:
4583:
4582:
4520:
4503:
4496:
4489:
4474:
4470:
4469:
4465:
4457:
4454:
4453:
4414:
4401:
4394:
4389:
4388:
4362:
4353:
4345:
4340:
4336:
4335:
4323:
4308:
4304:
4303:
4299:
4291:
4288:
4287:
4255:
4244:
4234:
4213:, at any point
4212:
4139:
4120:
4074:
4070:
4064:
4060:
4051:
4040:
4034:
4031:
4030:
4020:
3987:
3979:
3969:
3965:
3959:
3955:
3946:
3935:
3914:
3911:
3910:
3864:
3562:
3558:
3531:
3527:
3515:
3496:
3492:
3459:
3455:
3428:
3424:
3412:
3394:
3390:
3388:
3373:
3369:
3361:
3358:
3357:
3332:
3259: is the
3253:
3161:
3157:
3145:
3141:
3139:
3136:
3135:
3121:
3110:
3063:
3059:
3055:
3046:
3042:
3032:
3028:
3010:
2991:
2987:
2970:
2967:
2966:
2952:
2934:
2855:
2848:
2841:
2834:
2823:
2813:
2724:
2720:
2703:
2699:
2692:
2686:
2683:
2682:
2637:
2633:
2629:
2604:
2600:
2596:
2592:
2588:
2584:
2576:
2572:
2565:
2559:
2556:
2555:
2520:
2517:
2516:
2488:
2485:
2484:
2468:
2465:
2464:
2429:
2410:
2406:
2402:
2397:
2393:
2371:
2367:
2366:
2350:
2346:
2345:
2343:
2337:
2329:
2315:
2310:
2309:
2305:
2278:
2275:
2274:
2209:
2176:in the sense of
2172:
2153:
2074:
2046:
2042:
2021:
2002:
1998:
1990:
1987:
1986:
1946:
1938:
1921:
1910:
1908:
1891:
1888:
1887:
1827:
1765:
1757:
1739:
1735:
1729:
1725:
1719:
1708:
1687:
1684:
1683:
1658:
1657:
1655:
1652:
1651:
1636:
1630:
1619:
1593:
1592:
1590:
1587:
1586:
1552:
1551:
1549:
1546:
1545:
1515:
1514:
1512:
1509:
1508:
1462:
1448:
1444:
1440:
1429:
1428:
1407:
1403:
1394:
1389:
1375:
1370:
1353:
1349:
1345:
1341:
1336:
1333:
1332:
1326:
1288:
1279:
1275:
1261:
1260:
1258:
1255:
1254:
1186:
1170:
1166:
1153:
1149:
1145:
1133:
1128:
1114:
1109:
1086:
1085:
1068:
1057:
1054:
1053:
1003:
977:
976:
973: and
971:
962:
958:
952:
948:
937:
928:
924:
918:
914:
906:
903:
902:
896:Katznelson 1976
861:
860:
858:
855:
854:
809:
805:
804:
800:
783:
779:
778:
774:
763:
762:
757:
754:
753:
745:belongs to the
724:
723:
721:
718:
717:
689:
685:
681:
677:
662:
642:
638:
634:
623:
622:
620:
617:
616:
598:Katznelson 1976
587:
511:
507:
494:
486:
485:
470:
466:
465:
461:
453:
450:
449:
409:
405:
404:
400:
392:
389:
388:
342:
329:
322:
303:
299:
295:
291:
287:
283:
282:
273:
268:
254:
249:
248:
244:
243:
225:
219:
216:
215:
165:For spaces of
163:
24:
17:
12:
11:
5:
7678:
7668:
7667:
7662:
7657:
7652:
7635:
7634:
7632:
7631:
7620:
7617:
7616:
7614:
7613:
7608:
7603:
7598:
7596:Choquet theory
7593:
7588:
7582:
7580:
7576:
7575:
7573:
7572:
7562:
7557:
7552:
7547:
7542:
7537:
7532:
7527:
7522:
7517:
7512:
7506:
7504:
7500:
7499:
7497:
7496:
7491:
7485:
7483:
7479:
7478:
7476:
7475:
7470:
7465:
7460:
7455:
7450:
7448:Banach algebra
7444:
7442:
7438:
7437:
7435:
7434:
7429:
7424:
7419:
7414:
7409:
7404:
7399:
7394:
7389:
7383:
7381:
7377:
7376:
7374:
7373:
7371:BanachâAlaoglu
7368:
7363:
7358:
7353:
7348:
7343:
7338:
7333:
7327:
7325:
7319:
7318:
7315:
7314:
7312:
7311:
7306:
7301:
7299:Locally convex
7296:
7282:
7277:
7271:
7269:
7265:
7264:
7262:
7261:
7256:
7251:
7246:
7241:
7236:
7231:
7226:
7221:
7216:
7210:
7204:
7200:
7199:
7183:
7182:
7175:
7168:
7160:
7154:
7153:
7135:
7130:
7110:
7077:
7064:
7043:
7038:
7020:
7009:
7001:
6996:
6976:
6971:
6954:
6917:
6908:W. A. Benjamin
6899:
6886:"Hardy spaces"
6881:
6837:
6834:Academic Press
6822:
6817:
6800:
6795:
6778:
6725:
6722:
6720:
6719:
6680:
6644:
6642:
6639:
6615:
6614:
6603:
6600:
6596:
6588:
6585:
6578:
6570:
6565:
6560:
6557:
6554:
6549:
6545:
6539:
6535:
6530:
6526:
6521:
6514:
6509:
6505:
6478:
6477:
6466:
6461:
6457:
6451:
6447:
6443:
6440:
6437:
6421:
6401:
6397:
6391:
6371:
6370:
6359:
6354:
6351:
6348:
6344:
6340:
6337:
6334:
6329:
6325:
6296:
6264:
6263:
6252:
6246:
6243:
6237:
6231:
6226:
6219:
6215:
6211:
6206:
6203:
6200:
6196:
6191:
6185:
6180:
6177:
6174:
6170:
6166:
6161:
6156:
6149:
6145:
6140:
6135:
6130:
6127:
6124:
6121:
6118:
6075:) when 1 <
6051:
6050:
6037:
6030:
6026:
6021:
6017:
6012:
6007:
6004:
6001:
5996:
5992:
5958:
5949:
5930:
5915:is bounded in
5909:
5904:
5868:
5863:
5854:
5853:
5842:
5838:
5832:
5828:
5823:
5816:
5813:
5810:
5806:
5802:
5797:
5793:
5770:
5764:
5757:
5751:
5736:
5731:
5725:
5719:
5683:Haar functions
5678:
5674:
5662:
5651:
5646:
5645:
5634:
5631:
5626:
5621:
5614:
5610:
5605:
5601:
5589:
5584:
5580:
5574:
5570:
5566:
5563:
5560:
5456:, 0 <
5432:
5426:
5420:
5419:
5408:
5405:
5401:
5392:
5388:
5382:
5378:
5374:
5367:
5363:
5357:
5353:
5349:
5346:
5343:
5340:
5333:
5328:
5322:
5294:
5291:
5272:
5190:
5189:
5178:
5175:
5172:
5160:
5157:
5154:
5151:
5148:
5145:
5140:
5137:
5134:
5131:
5128:
5123:
5118:
5115:
5112:
5109:
5106:
5103:
5098:
5095:
5092:
5089:
5086:
5081:
5076:
5073:
5070:
5067:
5062:
5058:
4988:
4967:
4928:
4915:
4914:
4902:
4898:
4895:
4892:
4889:
4884:
4880:
4876:
4873:
4870:
4866:
4860:
4857:
4854:
4850:
4846:
4843:
4840:
4837:
4834:
4831:
4826:
4822:
4818:
4777:
4771:
4760:
4759:
4748:
4745:
4742:
4739:
4736:
4733:
4730:
4727:
4721:
4718:
4715:
4710:
4704:
4701:
4698:
4695:
4692:
4689:
4686:
4683:
4649:
4648:
4637:
4631:
4628:
4625:
4620:
4617:
4614:
4608:
4605:
4602:
4599:
4596:
4593:
4590:
4539:
4538:
4527:
4523:
4519:
4516:
4513:
4510:
4506:
4499:
4495:
4492:
4488:
4484:
4477:
4473:
4468:
4464:
4461:
4439:
4438:
4427:
4421:
4418:
4412:
4408:
4404:
4397:
4392:
4387:
4384:
4381:
4378:
4375:
4372:
4369:
4365:
4359:
4356:
4351:
4348:
4344:
4339:
4332:
4329:
4326:
4322:
4318:
4311:
4307:
4302:
4298:
4295:
4254:
4251:
4240:
4230:
4208:
4137:
4118:
4097:
4096:
4083:
4080:
4077:
4073:
4067:
4063:
4057:
4054:
4049:
4046:
4043:
4039:
4018:
4009:
4008:
3997:
3994:
3990:
3986:
3982:
3977:
3972:
3968:
3962:
3958:
3952:
3949:
3944:
3941:
3938:
3934:
3930:
3927:
3924:
3921:
3918:
3863:
3856:
3855:
3854:
3835:
3799: of an
3722:
3721:
3706:
3687:
3600:
3599:
3588:
3585:
3582:
3579:
3576:
3573:
3570:
3565:
3561:
3557:
3554:
3551:
3548:
3545:
3542:
3539:
3534:
3530:
3524:
3521:
3518:
3514:
3510:
3507:
3502:
3499:
3495:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3462:
3458:
3454:
3451:
3448:
3445:
3442:
3439:
3436:
3431:
3427:
3421:
3418:
3415:
3411:
3407:
3402:
3397:
3393:
3387:
3384:
3379:
3376:
3372:
3368:
3365:
3331:
3328:
3251:
3221: is in
3196:
3195:
3184:
3181:
3178:
3175:
3172:
3169:
3164:
3160:
3156:
3151:
3148:
3144:
3119:
3108:
3095:
3094:
3083:
3079:
3074:
3069:
3066:
3062:
3058:
3054:
3049:
3045:
3041:
3038:
3035:
3031:
3025:
3022:
3019:
3016:
3013:
3009:
3005:
3002:
2997:
2994:
2990:
2986:
2983:
2980:
2977:
2974:
2950:
2933:
2930:
2862:
2861:
2853:
2846:
2839:
2832:
2821:
2811:
2747:
2746:
2735:
2730:
2727:
2723:
2719:
2716:
2713:
2706:
2702:
2698:
2695:
2691:
2666:One says that
2661:
2660:
2648:
2643:
2640:
2636:
2632:
2628:
2625:
2621:
2616:
2610:
2607:
2603:
2599:
2595:
2591:
2587:
2579:
2575:
2571:
2568:
2564:
2545:Poisson kernel
2524:
2504:
2501:
2498:
2495:
2492:
2472:
2453:
2452:
2440:
2436:
2432:
2426:
2421:
2416:
2413:
2409:
2405:
2400:
2396:
2391:
2385:
2382:
2377:
2374:
2370:
2364:
2361:
2356:
2353:
2349:
2340:
2335:
2332:
2328:
2321:
2318:
2314:
2308:
2304:
2301:
2297:
2294:
2291:
2288:
2285:
2282:
2258:One says that
2245:inner function
2237:outer function
2208:
2205:
2170:
2151:
2103:
2102:
2091:
2088:
2082:
2079:
2073:
2070:
2067:
2063:
2060:
2057:
2052:
2049:
2045:
2041:
2038:
2035:
2030:
2027:
2024:
2020:
2016:
2013:
2008:
2005:
2001:
1997:
1994:
1968:
1967:
1956:
1953:
1949:
1945:
1941:
1936:
1930:
1927:
1924:
1919:
1916:
1913:
1907:
1904:
1901:
1898:
1895:
1826:
1823:
1787:
1786:
1775:
1772:
1768:
1764:
1760:
1747:
1742:
1738:
1732:
1728:
1722:
1717:
1714:
1711:
1707:
1703:
1700:
1697:
1694:
1691:
1665:
1662:
1650:associated to
1638:= 0 for every
1634:
1622:
1617:
1600:
1597:
1559:
1556:
1522:
1519:
1497:
1496:
1485:
1482:
1479:
1476:
1472:
1469:
1465:
1459:
1454:
1451:
1447:
1443:
1436:
1433:
1427:
1424:
1421:
1418:
1415:
1410:
1406:
1400:
1397:
1392:
1388:
1381:
1378:
1374:
1369:
1365:
1359:
1356:
1352:
1348:
1344:
1340:
1324:
1320:Poisson kernel
1295:
1291:
1287:
1282:
1278:
1274:
1268:
1265:
1208:
1207:
1196:
1193:
1189:
1182:
1179:
1176:
1173:
1169:
1164:
1159:
1156:
1152:
1148:
1144:
1139:
1136:
1131:
1127:
1120:
1117:
1113:
1108:
1105:
1102:
1099:
1093:
1090:
1075:
1071:
1067:
1064:
1061:
1045:of a function
1031:
1030:
1019:
1016:
1013:
1010:
1002:
999:
996:
993:
990:
984:
981:
969:
965:
961:
955:
951:
947:
944:
935:
931:
927:
921:
917:
913:
910:
868:
865:
831:
830:
819:
812:
808:
803:
799:
796:
793:
786:
782:
777:
770:
767:
761:
731:
728:
714:
713:
701:
695:
692:
688:
684:
680:
676:
671:
668:
665:
661:
657:
653:
648:
645:
641:
637:
630:
627:
586:
583:
543:
542:
531:
527:
523:
520:
517:
514:
510:
504:
501:
497:
493:
489:
484:
480:
473:
469:
464:
460:
457:
419:
412:
408:
403:
399:
396:
373:
372:
361:
358:
355:
349:
346:
340:
336:
332:
325:
320:
315:
309:
306:
302:
298:
294:
290:
286:
279:
276:
271:
267:
260:
257:
253:
247:
240:
237:
234:
231:
228:
224:
162:
159:
144:control theory
15:
9:
6:
4:
3:
2:
7677:
7666:
7663:
7661:
7658:
7656:
7653:
7651:
7648:
7647:
7645:
7630:
7622:
7621:
7618:
7612:
7609:
7607:
7604:
7602:
7601:Weak topology
7599:
7597:
7594:
7592:
7589:
7587:
7584:
7583:
7581:
7577:
7570:
7566:
7563:
7561:
7558:
7556:
7553:
7551:
7548:
7546:
7543:
7541:
7538:
7536:
7533:
7531:
7528:
7526:
7525:Index theorem
7523:
7521:
7518:
7516:
7513:
7511:
7508:
7507:
7505:
7501:
7495:
7492:
7490:
7487:
7486:
7484:
7482:Open problems
7480:
7474:
7471:
7469:
7466:
7464:
7461:
7459:
7456:
7454:
7451:
7449:
7446:
7445:
7443:
7439:
7433:
7430:
7428:
7425:
7423:
7420:
7418:
7415:
7413:
7410:
7408:
7405:
7403:
7400:
7398:
7395:
7393:
7390:
7388:
7385:
7384:
7382:
7378:
7372:
7369:
7367:
7364:
7362:
7359:
7357:
7354:
7352:
7349:
7347:
7344:
7342:
7339:
7337:
7334:
7332:
7329:
7328:
7326:
7324:
7320:
7310:
7307:
7305:
7302:
7300:
7297:
7294:
7290:
7286:
7283:
7281:
7278:
7276:
7273:
7272:
7270:
7266:
7260:
7257:
7255:
7252:
7250:
7247:
7245:
7242:
7240:
7237:
7235:
7232:
7230:
7227:
7225:
7222:
7220:
7217:
7215:
7212:
7211:
7208:
7205:
7201:
7196:
7192:
7188:
7181:
7176:
7174:
7169:
7167:
7162:
7161:
7158:
7151:
7147:
7146:
7141:
7136:
7133:
7127:
7123:
7119:
7115:
7114:Rudin, Walter
7111:
7108:
7104:
7100:
7096:
7092:
7088:
7087:
7082:
7078:
7075:
7071:
7067:
7061:
7057:
7053:
7049:
7044:
7041:
7039:9780521517683
7035:
7031:
7030:
7025:
7024:Mashreghi, J.
7021:
7018:
7014:
7007:
7002:
6999:
6993:
6989:
6985:
6981:
6977:
6974:
6968:
6964:
6960:
6955:
6952:
6948:
6944:
6940:
6936:
6932:
6931:
6926:
6922:
6918:
6916:
6913:
6909:
6905:
6900:
6897:
6893:
6892:
6887:
6882:
6878:
6874:
6869:
6864:
6860:
6856:
6855:
6850:
6846:
6842:
6838:
6835:
6831:
6827:
6823:
6820:
6814:
6810:
6807:, Ann Arbor:
6806:
6801:
6798:
6792:
6788:
6784:
6779:
6775:
6771:
6767:
6763:
6759:
6755:
6751:
6747:
6743:
6739:
6738:
6733:
6728:
6727:
6714:
6709:
6705:
6701:
6700:
6695:
6691:
6684:
6675:
6670:
6666:
6662:
6661:
6656:
6649:
6645:
6638:
6636:
6632:
6628:
6624:
6620:
6601:
6598:
6586:
6583:
6568:
6555:
6547:
6543:
6537:
6533:
6524:
6512:
6507:
6503:
6495:
6494:
6493:
6491:
6487:
6483:
6464:
6459:
6455:
6449:
6445:
6441:
6438:
6435:
6428:
6427:
6426:
6424:
6417:
6413:
6409:
6404:
6396:
6390:
6388:
6384:
6381: â
6380:
6376:
6352:
6349:
6346:
6342:
6335:
6332:
6327:
6323:
6315:
6314:
6313:
6311:
6307:
6303:
6299:
6292:
6287:
6285:
6281:
6277:
6273:
6269:
6250:
6244:
6241:
6235:
6229:
6217:
6213:
6209:
6204:
6201:
6198:
6194:
6178:
6175:
6172:
6168:
6164:
6159:
6147:
6143:
6133:
6128:
6122:
6116:
6109:
6108:
6107:
6105:
6101:
6097:
6093:
6088:
6086:
6082:
6078:
6074:
6070:
6066:
6062:
6058:
6056:
6028:
6015:
6005:
5999:
5994:
5990:
5982:
5981:
5980:
5978:
5974:
5970:
5966:
5961:
5956:
5952:
5945:
5942:-norm by the
5941:
5937:
5934:converges to
5933:
5926:
5922:
5918:
5912:
5907:
5900:
5896:
5891:
5889:
5885:
5881:
5877:
5871:
5866:
5859:
5840:
5830:
5826:
5814:
5811:
5808:
5800:
5795:
5791:
5783:
5782:
5781:
5779:
5773:
5767:
5760:
5754:
5749:
5745:
5739:
5734:
5724:
5718:
5716:
5712:
5708:
5704:
5700:
5696:
5692:
5688:
5684:
5672:
5667:
5666:are scalars.
5665:
5658:
5654:
5629:
5624:
5612:
5608:
5599:
5587:
5582:
5578:
5572:
5568:
5564:
5561:
5558:
5551:
5550:
5549:
5547:
5543:
5539:
5535:
5531:
5526:
5524:
5520:
5516:
5512:
5508:
5504:
5500:
5498:
5494:is called an
5493:
5489:
5485:
5481:
5476:
5472:
5468:
5463:
5459:
5455:
5451:
5447:
5443:
5439:
5435:
5425:
5406:
5403:
5390:
5386:
5380:
5376:
5372:
5365:
5361:
5355:
5351:
5344:
5338:
5331:
5320:
5312:
5311:
5310:
5308:
5304:
5300:
5290:
5288:
5284:
5280:
5275:
5270:
5266:
5262:
5258:
5254:
5250:
5246:
5242:
5237:
5235:
5231:
5227:
5224:. The space
5223:
5219:
5216:is the space
5215:
5211:
5207:
5203:
5199:
5195:
5176:
5173:
5170:
5158:
5152:
5149:
5146:
5135:
5132:
5129:
5116:
5110:
5107:
5104:
5093:
5090:
5087:
5074:
5068:
5060:
5056:
5048:
5047:
5046:
5044:
5040:
5036:
5032:
5028:
5024:
5020:
5016:
5012:
5007:
5005:
5001:
4997:
4993:
4987:
4983:
4979:
4975:
4970:
4965:
4961:
4957:
4951:
4947:
4943:
4937:
4931:
4924:
4920:
4893:
4882:
4874:
4871:
4858:
4855:
4852:
4844:
4838:
4829:
4820:
4809:
4808:
4807:
4806:
4802:
4798:
4795:
4791:
4787:
4783:
4776:
4770:
4768:
4765:
4746:
4737:
4731:
4725:
4719:
4716:
4713:
4708:
4702:
4696:
4687:
4684:
4674:
4673:
4672:
4671:) defined by
4670:
4666:
4662:
4658:
4654:
4635:
4629:
4626:
4623:
4618:
4615:
4612:
4606:
4603:
4600:
4594:
4588:
4581:
4580:
4579:
4577:
4573:
4569:
4565:
4564:
4559:
4555:
4551:
4547:
4544:
4541:Although the
4525:
4514:
4508:
4493:
4490:
4482:
4471:
4462:
4452:
4451:
4450:
4448:
4444:
4425:
4419:
4416:
4410:
4406:
4395:
4382:
4379:
4376:
4373:
4367:
4354:
4346:
4342:
4337:
4330:
4327:
4324:
4316:
4309:
4305:
4296:
4286:
4285:
4284:
4282:
4278:
4274:
4271:
4267:
4263:
4258:
4250:
4248:
4243:
4238:
4233:
4228:
4224:
4220:
4216:
4211:
4206:
4202:
4198:
4194:
4189:
4187:
4183:
4179:
4175:
4171:
4167:
4163:
4159:
4155:
4150:
4148:
4144:
4140:
4133:
4129:
4125:
4121:
4115: â
4114:
4110:
4106:
4102:
4081:
4078:
4075:
4071:
4065:
4061:
4052:
4047:
4044:
4041:
4037:
4029:
4028:
4027:
4025:
4021:
4014:
3995:
3992:
3984:
3975:
3970:
3966:
3960:
3956:
3947:
3942:
3939:
3936:
3932:
3928:
3922:
3916:
3909:
3908:
3907:
3905:
3901:
3897:
3893:
3889:
3885:
3881:
3877:
3873:
3869:
3861:
3852:
3848:
3844:
3840:
3837:the function
3836:
3833:
3829:
3825:
3821:
3818:the function
3817:
3816:
3815:
3814:
3810:
3806:
3802:
3798:
3794:
3789:
3787:
3783:
3779:
3775:
3771:
3767:
3763:
3759:
3755:
3751:
3747:
3743:
3739:
3735:
3731:
3727:
3719:
3715:
3711:
3707:
3704:
3700:
3696:
3692:
3689:the function
3688:
3685:
3681:
3677:
3673:
3670:the function
3669:
3668:
3667:
3665:
3661:
3657:
3653:
3651:
3647:
3641:
3637:
3633:
3629:
3625:
3621:
3617:
3613:
3609:
3605:
3602:This mapping
3586:
3580:
3577:
3571:
3568:
3563:
3559:
3555:
3549:
3546:
3540:
3537:
3532:
3528:
3522:
3519:
3516:
3512:
3508:
3500:
3497:
3493:
3486:
3477:
3474:
3468:
3465:
3460:
3456:
3452:
3446:
3443:
3437:
3434:
3429:
3425:
3419:
3416:
3413:
3409:
3405:
3400:
3395:
3391:
3385:
3377:
3374:
3370:
3363:
3356:
3355:
3354:
3352:
3348:
3344:
3341:
3337:
3327:
3325:
3321:
3317:
3313:
3310:
3306:
3302:
3298:
3294:
3290:
3286:
3282:
3278:
3274:
3270:
3266:
3262:
3258:
3254:
3247:
3243:
3239:
3235:
3232:The function
3230:
3228:
3224:
3220:
3216:
3212:
3208:
3205:
3201:
3182:
3176:
3173:
3170:
3162:
3158:
3149:
3146:
3142:
3134:
3133:
3132:
3130:
3126:
3122:
3115:
3111:
3104:
3100:
3081:
3077:
3072:
3067:
3064:
3060:
3056:
3047:
3043:
3039:
3036:
3029:
3023:
3020:
3017:
3014:
3011:
3003:
2995:
2992:
2988:
2978:
2975:
2965:
2964:
2963:
2961:
2957:
2953:
2945:
2943:
2939:
2929:
2927:
2923:
2919:
2915:
2911:
2907:
2903:
2899:
2895:
2891:
2887:
2883:
2879:
2876:< â and 1/
2875:
2871:
2867:
2859:
2852:
2845:
2838:
2831:
2827:
2820:
2817:
2816:
2815:
2810:
2806:
2801:
2799:
2795:
2791:
2787:
2783:
2779:
2775:
2772:The function
2770:
2768:
2764:
2760:
2756:
2752:
2728:
2725:
2721:
2717:
2711:
2704:
2700:
2693:
2681:
2680:
2679:
2677:
2673:
2669:
2664:
2646:
2641:
2638:
2634:
2630:
2626:
2623:
2619:
2614:
2608:
2605:
2601:
2597:
2593:
2589:
2585:
2577:
2573:
2566:
2554:
2553:
2552:
2550:
2546:
2542:
2538:
2522:
2499:
2493:
2490:
2470:
2462:
2458:
2438:
2434:
2424:
2419:
2414:
2411:
2407:
2403:
2398:
2394:
2389:
2383:
2380:
2375:
2372:
2368:
2362:
2359:
2354:
2351:
2347:
2338:
2333:
2330:
2326:
2319:
2316:
2312:
2306:
2302:
2299:
2295:
2292:
2286:
2280:
2273:
2272:
2271:
2269:
2265:
2261:
2256:
2254:
2250:
2246:
2242:
2238:
2234:
2230:
2226:
2222:
2218:
2214:
2204:
2202:
2198:
2194:
2190:
2186:
2182:
2181:
2180:distributions
2177:
2173:
2166:
2162:
2158:
2154:
2147:
2142:
2140:
2136:
2132:
2128:
2124:
2120:
2116:
2112:
2108:
2089:
2080:
2077:
2068:
2065:
2061:
2058:
2050:
2047:
2043:
2039:
2033:
2028:
2022:
2014:
2006:
2003:
1999:
1992:
1985:
1984:
1983:
1981:
1977:
1973:
1954:
1951:
1943:
1934:
1928:
1925:
1922:
1917:
1914:
1911:
1905:
1899:
1893:
1886:
1885:
1884:
1882:
1877:
1875:
1871:
1867:
1863:
1859:
1855:
1851:
1847:
1843:
1839:
1836:
1832:
1822:
1820:
1816:
1812:
1808:
1804:
1800:
1796:
1792:
1773:
1770:
1762:
1745:
1740:
1736:
1730:
1726:
1715:
1712:
1709:
1705:
1701:
1695:
1689:
1682:
1681:
1680:
1660:
1649:
1645:
1641:
1637:
1629:
1625:
1620:
1595:
1584:
1580:
1576:
1554:
1543:
1539:
1517:
1507:exactly when
1506:
1502:
1483:
1480:
1477:
1474:
1470:
1467:
1457:
1452:
1449:
1445:
1441:
1431:
1422:
1419:
1416:
1408:
1404:
1398:
1395:
1390:
1386:
1379:
1376:
1372:
1367:
1363:
1357:
1354:
1350:
1346:
1342:
1338:
1331:
1330:
1329:
1327:
1321:
1317:
1313:
1309:
1280:
1276:
1272:
1263:
1251:
1249:
1245:
1241:
1237:
1233:
1229:
1225:
1221:
1217:
1213:
1194:
1191:
1180:
1177:
1174:
1171:
1167:
1162:
1157:
1154:
1150:
1146:
1142:
1137:
1134:
1129:
1125:
1118:
1115:
1111:
1106:
1100:
1088:
1073:
1065:
1062:
1052:
1051:
1050:
1048:
1044:
1040:
1036:
1017:
1014:
1011:
1008:
1000:
997:
991:
979:
967:
959:
953:
949:
945:
942:
933:
925:
919:
915:
911:
908:
901:
900:
899:
897:
893:
889:
885:
863:
852:
848:
844:
840:
836:
817:
810:
806:
797:
791:
784:
780:
765:
752:
751:
750:
748:
726:
699:
693:
690:
686:
682:
678:
674:
669:
663:
655:
651:
646:
643:
639:
635:
625:
615:
614:
613:
611:
607:
603:
599:
595:
593:
582:
580:
576:
572:
568:
564:
560:
556:
552:
548:
529:
525:
518:
512:
508:
502:
499:
491:
478:
467:
458:
448:
447:
446:
444:
439:
437:
433:
417:
410:
406:
397:
387:, denoted by
386:
382:
378:
359:
353:
347:
344:
338:
334:
323:
318:
313:
307:
304:
300:
296:
292:
288:
284:
277:
274:
269:
265:
258:
255:
251:
245:
238:
235:
232:
229:
226:
214:
213:
212:
210:
206:
202:
197:
195:
191:
187:
183:
179:
178:
172:
168:
158:
156:
152:
150:
145:
141:
136:
134:
130:
125:
123:
119:
115:
111:
107:
103:
99:
95:
93:
88:
84:
83:distributions
80:
77:
76:real analysis
73:
69:
65:
61:
60:Frigyes Riesz
57:
53:
49:
45:
41:
37:
36:Hardy classes
33:
29:
22:
7650:Hardy spaces
7591:Balanced set
7565:Distribution
7509:
7503:Applications
7356:KreinâMilman
7341:Closed graph
7143:
7117:
7090:
7084:
7051:
7047:
7028:
7012:
7005:
6983:
6958:
6934:
6928:
6921:Hardy, G. H.
6903:
6889:
6858:
6852:
6848:
6829:
6804:
6782:
6741:
6735:
6731:
6703:
6697:
6683:
6664:
6658:
6648:
6634:
6622:
6618:
6616:
6489:
6485:
6481:
6479:
6419:
6411:
6407:
6402:
6399:
6394:
6382:
6378:
6374:
6372:
6309:
6305:
6304:= 0 at time
6301:
6294:
6288:
6279:
6275:
6271:
6267:
6265:
6103:
6099:
6095:
6091:
6089:
6080:
6076:
6072:
6068:
6064:
6059:
6054:
6052:
5976:
5972:
5968:
5964:
5959:
5954:
5947:
5939:
5935:
5928:
5927:. Moreover,
5920:
5916:
5910:
5902:
5898:
5894:
5892:
5887:
5883:
5879:
5875:
5869:
5861:
5857:
5855:
5777:
5771:
5765:
5758:
5752:
5747:
5737:
5729:
5727:
5722:
5714:
5710:
5706:
5702:
5698:
5694:
5690:
5686:
5670:
5668:
5660:
5656:
5649:
5647:
5545:
5541:
5537:
5533:
5529:
5528:When 0 <
5527:
5522:
5518:
5514:
5510:
5506:
5502:
5496:
5495:
5491:
5487:
5483:
5479:
5477:
5470:
5466:
5461:
5457:
5453:
5449:
5445:
5441:
5437:
5430:
5423:
5422:whose order
5421:
5306:
5302:
5298:
5297:When 0 <
5296:
5286:
5282:
5278:
5273:
5268:
5264:
5260:
5259:< 1, the
5256:
5252:
5248:
5244:
5240:
5238:
5233:
5229:
5225:
5217:
5213:
5209:
5205:
5201:
5197:
5193:
5191:
5042:
5038:
5034:
5030:
5026:
5022:
5018:
5014:
5010:
5008:
5003:
4999:
4995:
4991:
4985:
4981:
4977:
4973:
4968:
4963:
4955:
4949:
4945:
4941:
4935:
4929:
4922:
4918:
4916:
4796:
4789:
4785:
4781:
4779:
4774:
4761:
4668:
4664:
4660:
4656:
4652:
4650:
4575:
4571:
4567:
4562:
4549:
4545:
4540:
4446:
4442:
4440:
4280:
4276:
4272:
4265:
4261:
4259:
4256:
4246:
4241:
4236:
4235:belong when
4231:
4226:
4225:) for every
4222:
4218:
4214:
4209:
4204:
4200:
4196:
4192:
4190:
4185:
4181:
4177:
4173:
4169:
4165:
4161:
4157:
4153:
4151:
4146:
4142:
4135:
4131:
4127:
4123:
4116:
4112:
4108:
4104:
4100:
4098:
4023:
4016:
4012:
4010:
3903:
3895:
3891:
3887:
3883:
3879:
3875:
3871:
3867:
3866:When 0 <
3865:
3859:
3850:
3846:
3842:
3838:
3831:
3827:
3823:
3819:
3812:
3808:
3807:be equal to
3804:
3800:
3796:
3792:
3791:The case of
3790:
3785:
3781:
3777:
3773:
3769:
3765:
3761:
3757:
3753:
3749:
3745:
3741:
3737:
3733:
3729:
3725:
3724:When 1 <
3723:
3717:
3713:
3709:
3702:
3698:
3694:
3690:
3683:
3679:
3675:
3671:
3663:
3659:
3655:
3649:
3645:
3639:
3635:
3631:
3623:
3619:
3615:
3611:
3607:
3603:
3601:
3350:
3346:
3342:
3339:
3335:
3333:
3323:
3319:
3315:
3311:
3308:
3304:
3300:
3296:
3292:
3288:
3284:
3280:
3276:
3272:
3268:
3264:
3260:
3256:
3249:
3245:
3241:
3237:
3233:
3231:
3226:
3222:
3218:
3214:
3210:
3206:
3203:
3199:
3197:
3128:
3124:
3117:
3113:
3106:
3102:
3098:
3096:
2959:
2955:
2948:
2946:
2941:
2937:
2935:
2925:
2921:
2917:
2913:
2909:
2905:
2901:
2897:
2893:
2889:
2885:
2881:
2877:
2873:
2869:
2865:
2863:
2857:
2850:
2849: and |
2843:
2836:
2829:
2825:
2818:
2808:
2804:
2802:
2797:
2793:
2789:
2785:
2781:
2777:
2773:
2771:
2762:
2758:
2748:
2675:
2671:
2667:
2665:
2662:
2540:
2536:
2460:
2456:
2454:
2267:
2263:
2259:
2257:
2244:
2240:
2236:
2232:
2228:
2224:
2220:
2216:
2212:
2210:
2200:
2199:) for every
2196:
2192:
2188:
2179:
2175:
2168:
2164:
2160:
2156:
2149:
2145:
2143:
2138:
2134:
2130:
2126:
2122:
2118:
2114:
2110:
2106:
2104:
1979:
1975:
1971:
1969:
1880:
1878:
1873:
1869:
1865:
1861:
1857:
1853:
1849:
1845:
1841:
1837:
1834:
1833:< â, the
1830:
1828:
1818:
1813:consists of
1810:
1806:
1798:
1794:
1788:
1647:
1643:
1639:
1632:
1627:
1623:
1615:
1582:
1578:
1574:
1541:
1537:
1504:
1500:
1498:
1322:
1315:
1307:
1252:
1247:
1243:
1242:†â), so is
1239:
1236:Banach space
1231:
1227:
1223:
1219:
1215:
1211:
1209:
1046:
1038:
1034:
1032:
891:
887:
883:
850:
846:
842:
838:
834:
832:
746:
715:
609:
605:
601:
591:
588:
570:
562:
558:
554:
546:
544:
442:
440:
435:
431:
384:
380:
376:
374:
208:
204:
200:
198:
193:
189:
181:
176:
175:Hardy space
169:on the open
164:
148:
137:
132:
129:tube domains
126:
121:
117:
116:, while for
113:
108:are certain
105:
101:
91:
79:Hardy spaces
78:
42:are certain
39:
35:
32:Hardy spaces
31:
25:
7520:Heat kernel
7510:Hardy space
7417:Trace class
7331:HahnâBanach
7293:Topological
7122:McGraw-Hill
6937:: 269â277,
6744:: 137â153,
6667:: 239â255.
6627:MĂŒller 2005
6416:Haar system
6284:Garsia 1973
6266:Martingale-
6085:Garsia 1973
5874:belongs to
5721:Martingale
5464: >
5436:is at most
5267:th power ||
4767:isomorphism
3900:subharmonic
3732:belongs to
3660:real valued
2940:defined on
2749:exists for
1879:For 0 <
1817:indexed by
1805:indexed by
1503:belongs to
1314:) function
375:This class
203:for 0 <
120:< 1 the
68:G. H. Hardy
7644:Categories
7453:C*-algebra
7268:Properties
7056:BirkhÀuser
6951:45.1331.03
6724:References
5876:martingale
5744:martingale
5648:where the
3902:for every
3845:belong to
3697:belong to
3634:also maps
3345:such that
3217:such that
3097:where the
2753:Ξ and its
2751:almost all
2549:Rudin 1987
2249:Rudin 1987
2167:â 1 of Re(
2117:), but Re(
2109:and is in
1809:; whereas
1210:The space
1041:) are the
1033:where the
886:varies in
557:, and the
441:The space
383:-norm for
72:Hardy 1915
64:Riesz 1923
7427:Unbounded
7422:Transpose
7380:Operators
7309:Separable
7304:Reflexive
7289:Algebraic
7275:Barrelled
7150:EMS Press
7107:121306447
7093:: 87â95,
7081:Riesz, F.
6896:EMS Press
6847:(1972), "
6826:Duren, P.
6525:∑
6504:∫
6442:∑
6353:τ
6350:∧
6210:−
6184:∞
6169:∑
6025:Σ
6006:
5812:≥
5796:∗
5713:for some
5673: = ÎŽ
5633:∞
5600:∑
5565:∑
5373:…
5321:∫
5271: ||
5117:−
5108:−
4966: ||
4960:quasinorm
4879:Φ
4875:∗
4825:Φ
4764:isometric
4717:−
4709:π
4627:−
4607:⋅
4543:unit disk
4494:∈
4476:∞
4467:‖
4460:‖
4358:∞
4350:∞
4347:−
4343:∫
4301:‖
4294:‖
4268:) on the
4111:) =(
4082:θ
4056:∞
4038:∑
3951:∞
3933:∑
3581:θ
3572:
3556:−
3550:θ
3541:
3520:⩾
3513:∑
3501:θ
3484:⟶
3478:θ
3469:
3447:θ
3438:
3417:⩾
3410:∑
3378:θ
3287: "
3177:φ
3174:−
3171:θ
3155:→
3150:φ
3068:θ
3040:∗
2996:θ
2729:θ
2705:−
2697:→
2642:θ
2627:φ
2609:θ
2578:−
2570:→
2523:φ
2500:φ
2494:
2471:φ
2435:θ
2415:θ
2399:φ
2381:−
2376:θ
2355:θ
2339:π
2334:π
2331:−
2327:∫
2320:π
2303:
2078:θ
2069:
2051:θ
2026:→
2007:θ
1926:−
1868:) and Im(
1829:When 1 â€
1721:∞
1706:∑
1664:~
1599:~
1558:~
1521:~
1468:ϕ
1453:ϕ
1435:~
1423:ϕ
1420:−
1417:θ
1399:π
1387:∫
1380:π
1358:θ
1273:∈
1267:~
1238:(for 1 â€
1226:). Since
1192:ϕ
1181:ϕ
1172:−
1158:ϕ
1138:π
1126:∫
1119:π
1092:^
1066:∈
1060:∀
983:^
946:∈
912:∈
867:~
837:, and by
802:‖
795:‖
776:‖
769:~
760:‖
730:~
694:θ
667:→
647:θ
629:~
569:that the
472:∞
463:‖
456:‖
402:‖
395:‖
357:∞
335:θ
308:θ
278:π
266:∫
259:π
230:⩽
171:unit disk
153:) and in
146:(such as
52:unit disk
7629:Category
7441:Algebras
7323:Theorems
7280:Complete
7249:Schwartz
7195:glossary
7116:(1987),
7026:(2009),
6982:(1976),
6923:(1915),
6828:(1970),
6774:53996980
6692:(1965).
6484:norm of
5946:; hence
5856:Let 1 â€
5548:-atoms,
5513:). The
4984:norm of
4655: :
4570: :
4168:) (for |
3880:function
3797:M f
3728:< â,
3269:M f
3257:M f
3219:M f
2824: =
2784:, is in
2266:) is an
2253:Beurling
2125:from Re(
1312:harmonic
579:measures
7432:Unitary
7412:Nuclear
7397:Compact
7392:Bounded
7387:Adjoint
7361:Minâmax
7254:Sobolev
7239:Nuclear
7229:Hilbert
7224:Fréchet
7189: (
7074:2157745
6915:0448538
6877:0447953
6766:0274767
6758:1995838
5923:by the
5540:has an
5501:(here |
5490:| then
5429:+ ... +
4932:
4164:) = (1â
3740:) when
3267:. When
3127:on the
2755:modulus
1306:, with
1234:) is a
882:, when
608:, with
577:, i.e.
151:methods
110:subsets
87:complex
50:on the
7407:Normal
7244:Orlicz
7234:Hölder
7214:Banach
7203:Spaces
7191:topics
7128:
7105:
7072:
7062:
7052:spaces
7036:
7013:Spaces
6994:
6969:
6949:
6875:
6815:
6793:
6772:
6764:
6756:
5776:. The
5709:
5597:
5594:
5591:
5204:, and
5168:
5165:
5162:
4954:. The
4917:is in
4762:is an
4180:
4011:is in
3862:< 1
3303:. For
3279:) and
2842:
2761:is in
2670:is an
2539:is in
2459:with |
2243:is an
2235:is an
2231:where
2155:(e) =
2148:, let
1974:is in
1791:causal
1755:
1752:
1749:
1573:is in
1536:is in
1083:
1080:
1077:
594:spaces
551:subset
549:is a
184:whose
173:, the
94:spaces
74:). In
44:spaces
30:, the
7219:Besov
7103:S2CID
6770:S2CID
6754:JSTOR
6641:Notes
6637:(ÎŽ).
5882:when
5742:be a
5728:Let (
5701:when
5499:-atom
4245:when
3898:| is
3644:weak-
3642:) to
3244:) = (
2856:| = |
1970:Then
1631:with
1585:that
7567:(or
7285:Dual
7126:ISBN
7060:ISBN
7034:ISBN
6992:ISBN
6967:ISBN
6813:ISBN
6791:ISBN
6633:for
6377:iff
5957:on ÎŁ
5655:are
5630:<
5196:and
5192:The
5174:>
4856:>
4663:) â
4328:>
4022:= O(
3993:<
3843:H(f)
3730:H(f)
3695:H(f)
3099:star
3021:<
3015:<
2947:Let
2884:+ 1/
2880:= 1/
2803:Let
2239:and
1952:<
1771:<
1583:i.e.
1499:and
1478:<
1012:<
500:<
354:<
236:<
34:(or
7095:doi
6947:JFM
6939:doi
6863:doi
6859:129
6746:doi
6742:157
6734:",
6708:doi
6669:doi
6389:).
6286:).
6278:)â
6087:).
5938:in
5893:If
5805:sup
5536:of
5509:in
5473:+1)
5440:(1/
5251:(1/
5239:If
5236:).
5226:BMO
5218:BMO
4976:of
4938:) =
4849:sup
4487:sup
4321:sup
4279:on
4149:).
4141:of
3894:â |
3874:in
3788:).
3710:M f
3614:on
3569:cos
3538:sin
3466:sin
3435:cos
3349:+ i
3326:).
3263:of
3229:).
3008:sup
2892:in
2833:α+ÎČ
2690:lim
2563:lim
2491:log
2390:log
2300:exp
2219:in
2187:at
2174:),
2066:cot
2019:lim
1581:),
1250:).
660:lim
553:of
483:sup
223:sup
112:of
96:of
54:or
46:of
26:In
7646::
7193:â
7148:,
7142:,
7124:,
7120:,
7101:,
7091:18
7089:,
7070:MR
7068:,
7058:,
6990:,
6986:,
6965:,
6961:,
6945:,
6935:14
6933:,
6927:,
6912:MR
6894:,
6888:,
6873:MR
6871:,
6857:,
6843:;
6832:,
6811:,
6789:,
6785:,
6768:,
6762:MR
6760:,
6752:,
6740:,
6704:16
6702:.
6696:.
6665:81
6663:.
6657:.
6425:)
6057:.
5913:â„0
5897:â
5895:M*
5890:.
5886:â
5884:M*
5872:â„0
5774:â„0
5761:â„0
5740:â„0
5677:âÎŽ
5452:â
5274:Hp
5177:0.
4969:Hp
4962:||
4944:Ί(
4703::=
4574:â
4203:â
4126:â
4109:re
3853:).
3826:â
3744:â
3720:).
3678:â
3606:â
3289:is
3248:â
3242:re
3116:â
2924:,
2916:,
2872:,
2868:,
2835:=
2828:,
2800:.
2782:Gh
2780:=
2769:.
2229:Gh
2227:=
2161:re
2015::=
1955:1.
1821:.
1774:1.
1328::
898:)
604:â
157:.
38:)
7571:)
7295:)
7291:/
7287:(
7197:)
7179:e
7172:t
7165:v
7097::
7050:H
7010:p
7008:H
6941::
6879:.
6865::
6849:H
6776:.
6748::
6732:H
6716:.
6710::
6677:.
6671::
6635:H
6623:H
6619:H
6602:.
6599:x
6595:d
6587:2
6584:1
6577:)
6569:2
6564:|
6559:)
6556:x
6553:(
6548:k
6544:h
6538:k
6534:c
6529:|
6520:(
6513:1
6508:0
6490:L
6486:f
6482:H
6465:,
6460:k
6456:h
6450:k
6446:c
6439:=
6436:f
6422:k
6420:h
6418:(
6412:f
6408:n
6403:n
6395:H
6385:(
6383:H
6379:F
6375:H
6358:)
6347:t
6343:B
6339:(
6336:F
6333:=
6328:t
6324:M
6310:F
6306:t
6302:z
6297:t
6295:B
6293:(
6282:(
6280:L
6276:f
6274:(
6272:S
6268:H
6251:.
6245:2
6242:1
6236:)
6230:2
6225:|
6218:n
6214:M
6205:1
6202:+
6199:n
6195:M
6190:|
6179:0
6176:=
6173:n
6165:+
6160:2
6155:|
6148:0
6144:M
6139:|
6134:(
6129:=
6126:)
6123:f
6120:(
6117:S
6100:L
6096:p
6092:p
6081:H
6077:p
6073:P
6069:L
6065:H
6055:H
6036:)
6029:n
6020:|
6016:f
6011:(
6003:E
6000:=
5995:n
5991:M
5977:f
5973:P
5969:L
5965:H
5960:n
5955:f
5950:n
5948:M
5940:L
5936:f
5931:n
5929:M
5921:f
5917:L
5911:n
5908:)
5905:n
5903:M
5899:L
5888:L
5880:H
5878:-
5870:n
5867:)
5864:n
5862:M
5858:p
5841:.
5837:|
5831:n
5827:M
5822:|
5815:0
5809:n
5801:=
5792:M
5772:n
5769:)
5766:n
5759:n
5756:)
5753:n
5748:P
5738:n
5735:)
5732:n
5730:M
5723:H
5715:a
5711:x
5707:a
5703:p
5699:H
5695:p
5691:p
5687:H
5679:0
5675:1
5671:f
5663:j
5661:c
5657:H
5652:j
5650:a
5625:p
5620:|
5613:j
5609:c
5604:|
5588:,
5583:j
5579:a
5573:j
5569:c
5562:=
5559:f
5546:H
5538:H
5534:f
5530:p
5523:p
5519:H
5515:H
5511:R
5507:B
5503:B
5497:H
5492:f
5488:B
5484:B
5480:f
5471:n
5467:n
5462:p
5458:p
5454:H
5450:f
5446:f
5442:p
5438:n
5433:n
5431:i
5427:1
5424:i
5407:,
5404:x
5400:d
5391:n
5387:i
5381:n
5377:x
5366:1
5362:i
5356:1
5352:x
5348:)
5345:x
5342:(
5339:f
5332:n
5327:R
5307:H
5303:f
5299:p
5287:H
5283:H
5279:p
5269:f
5265:p
5261:H
5257:p
5253:p
5249:n
5245:H
5241:p
5234:L
5230:H
5214:H
5210:L
5206:H
5202:H
5198:H
5194:L
5171:k
5159:,
5156:)
5153:k
5150:+
5147:x
5144:(
5139:]
5136:1
5133:,
5130:0
5127:[
5122:1
5114:)
5111:k
5105:x
5102:(
5097:]
5094:1
5091:,
5088:0
5085:[
5080:1
5075:=
5072:)
5069:x
5066:(
5061:k
5057:f
5043:H
5039:L
5035:H
5031:L
5027:H
5023:p
5019:L
5015:H
5011:p
5004:p
5000:p
4996:H
4992:f
4989:Ί
4986:M
4982:L
4978:H
4974:f
4964:f
4958:-
4956:H
4952:)
4950:t
4946:x
4942:t
4936:x
4934:(
4930:t
4927:Ί
4923:R
4921:(
4919:L
4901:|
4897:)
4894:x
4891:(
4888:)
4883:t
4872:f
4869:(
4865:|
4859:0
4853:t
4845:=
4842:)
4839:x
4836:(
4833:)
4830:f
4821:M
4817:(
4797:f
4790:p
4786:H
4782:R
4775:R
4747:.
4744:)
4741:)
4738:z
4735:(
4732:m
4729:(
4726:f
4720:z
4714:1
4700:)
4697:z
4694:(
4691:)
4688:f
4685:M
4682:(
4669:D
4667:(
4665:H
4661:H
4659:(
4657:H
4653:M
4636:.
4630:z
4624:1
4619:z
4616:+
4613:1
4604:i
4601:=
4598:)
4595:z
4592:(
4589:m
4576:H
4572:D
4568:m
4563:H
4550:H
4546:D
4526:.
4522:|
4518:)
4515:z
4512:(
4509:f
4505:|
4498:H
4491:z
4483:=
4472:H
4463:f
4447:H
4445:(
4443:H
4426:.
4420:p
4417:1
4411:)
4407:x
4403:d
4396:p
4391:|
4386:)
4383:y
4380:i
4377:+
4374:x
4371:(
4368:f
4364:|
4355:+
4338:(
4331:0
4325:y
4317:=
4310:p
4306:H
4297:f
4281:H
4277:f
4273:H
4266:H
4264:(
4262:H
4247:p
4242:x
4237:p
4232:x
4227:p
4223:T
4221:(
4219:H
4215:x
4210:x
4205:H
4201:F
4197:T
4195:(
4193:H
4186:N
4182:p
4178:N
4174:H
4170:z
4166:z
4162:z
4160:(
4158:F
4154:p
4147:f
4143:F
4138:n
4136:c
4132:f
4128:H
4124:F
4119:r
4117:P
4113:f
4107:(
4105:F
4101:f
4079:n
4076:i
4072:e
4066:n
4062:c
4053:+
4048:0
4045:=
4042:n
4024:n
4019:n
4017:c
4013:H
3996:1
3989:|
3985:z
3981:|
3976:,
3971:n
3967:z
3961:n
3957:c
3948:+
3943:0
3940:=
3937:n
3929:=
3926:)
3923:z
3920:(
3917:F
3904:q
3896:z
3892:z
3884:L
3876:H
3872:F
3868:p
3860:p
3851:T
3849:(
3847:L
3839:f
3834:)
3832:T
3830:(
3828:H
3824:g
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