2279:
1980:
385:
2849:
1990:
1691:
1112:
706:
1256:
944:
1376:
2428:
2594:) theorem provides sufficient conditions for a singular integral operator to be a Calderón–Zygmund operator, that is for a singular integral operator associated to a Calderón–Zygmund kernel to be bounded on
1452:
799:
1533:
2554:
1681:
488:
251:
3155:
David; Semmes; Journé (1985). "Opérateurs de Calderón–Zygmund, fonctions para-accrétives et interpolation" (in French). Vol. 1. Revista Matemática
Iberoamericana. pp. 1–56.
2711:
2274:{\displaystyle |K(x,y)-K(x,y')|\leq {\frac {C|y-y'|^{\delta }}{{\bigl (}|x-y|+|x-y'|{\bigr )}^{n+\delta }}}{\text{ whenever }}|y-y'|\leq {\frac {1}{2}}\max {\bigl (}|x-y'|,|x-y|{\bigr )}}
1975:{\displaystyle |K(x,y)-K(x',y)|\leq {\frac {C|x-x'|^{\delta }}{{\bigl (}|x-y|+|x'-y|{\bigr )}^{n+\delta }}}{\text{ whenever }}|x-x'|\leq {\frac {1}{2}}\max {\bigl (}|x-y|,|x'-y|{\bigr )}}
127:
989:
2991:
3081:
3029:
196:| > ε as ε → 0, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on
526:
582:
1125:. Neither of the properties 1. or 2. is necessarily easy to verify, and a variety of sufficient conditions exist. Typically in applications, one also has a
188:| → 0. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over |
3485:
1135:
1546:
These are even more general operators. However, since our assumptions are so weak, it is not necessarily the case that these operators are bounded on
973:
553:
818:
17:
3544:
3105:
1275:
2308:
1538:
Observe that these conditions are satisfied for the
Hilbert and Riesz transforms, so this result is an extension of those result.
1394:
744:
1458:
2478:
1384:
The smoothness condition 2. is also often difficult to check in principle, the following sufficient condition of a kernel
39:
and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an
1603:
412:
380:{\displaystyle H(f)(x)={\frac {1}{\pi }}\lim _{\varepsilon \to 0}\int _{|x-y|>\varepsilon }{\frac {1}{x-y}}f(y)\,dy.}
2844:{\displaystyle \left|\int T{\bigl (}\tau ^{x}(\varphi _{r}){\bigr )}(y)\tau ^{x}(\psi _{r})(y)\,dy\right|\leq Cr^{-n}}
3503:
3474:
3447:
3303:
48:
3234:
1107:{\displaystyle \operatorname {p.v.} \,\,K=\lim _{\epsilon \to 0^{+}}\int _{|x|>\epsilon }\phi (x)K(x)\,dx}
3566:
2942:
567:
3298:, Cambridge Studies in Advanced Mathematics, vol. 48, Cambridge University Press, pp. xx+315,
3571:
3495:
3084:
3040:
3032:
149:
3576:
981:
2998:
3491:
701:{\displaystyle T(f)(x)=\lim _{\varepsilon \to 0}\int _{|y-x|>\varepsilon }K(x-y)f(y)\,dy.}
3513:
3457:
3401:
3349:
3313:
3271:
3209:
2922:
if it satisfies all of the following three conditions for some bounded accretive functions
504:
3521:
3465:
3409:
3321:
3279:
3217:
8:
3225:
3169:
1578:
3259:
1118:
3499:
3470:
3443:
3299:
3251:
3197:
733:
219:
213:
40:
36:
3368:, International Series of Monographs in Pure and Applied Mathematics, vol. 83,
3517:
3461:
3405:
3354:
3317:
3275:
3243:
3213:
3187:
3178:
1251:{\displaystyle \int _{R_{1}<|x|<R_{2}}K(x)\,dx=0,\ \forall R_{1},R_{2}>0}
3509:
3453:
3439:
3421:
3417:
3397:
3361:
3345:
3329:
3309:
3267:
3205:
391:
3393:
3287:
3229:
3173:
1582:
3536:
3560:
3435:
3381:
3255:
3201:
2441:
are smooth and have disjoint support. Such operators need not be bounded on
1266:
3333:
939:{\displaystyle \sup _{y\neq 0}\int _{|x|>2|y|}|K(x-y)-K(x)|\,dx\leq C.}
3481:
28:
3431:
3291:
3263:
3192:
2562:
It can be proved that such operators are, in fact, also bounded on all
3389:
3377:
3124:
Stein, Elias (1993). "Harmonic
Analysis". Princeton University Press.
2613:
supported in a ball of radius 1 and centred at the origin such that |
1371:{\displaystyle \sup _{R>0}\int _{R<|x|<2R}|K(x)|\,dx\leq C,}
390:
The most straightforward higher dimension analogues of these are the
3247:
1269:. If, in addition, one assumes 2. and the following size condition
2423:{\displaystyle \int g(x)T(f)(x)\,dx=\iint g(x)K(x,y)f(y)\,dy\,dx,}
1261:
which is quite easy to check. It is automatic, for instance, if
3487:
Singular integrals and differentiability properties of functions
2598:. In order to state the result we must first define some terms.
3427:
3373:
3369:
2286:
3385:
1541:
1587:
if it satisfies the following conditions for some constants
1447:{\displaystyle K\in C^{1}(\mathbf {R} ^{n}\setminus \{0\})}
794:{\displaystyle {\hat {K}}\in L^{\infty }(\mathbf {R} ^{n})}
3366:
Multidimensional singular integrals and integral equations
3176:(1952), "On the existence of certain singular integrals",
547:
1528:{\displaystyle |\nabla K(x)|\leq {\frac {C}{|x|^{n+1}}}}
3537:"Singular Integrals: The Roles of Calderón and Zygmund"
558:
A singular integral of convolution type is an operator
2918:
associated to a Calderón–Zygmund kernel is bounded on
3043:
3001:
2945:
2714:
2481:
2311:
1993:
1694:
1606:
1461:
1397:
1278:
1138:
992:
821:
747:
585:
507:
415:
254:
51:
3296:
Wavelets: Calderón-Zygmund and multilinear operators
2549:{\displaystyle \|T(f)\|_{L^{2}}\leq C\|f\|_{L^{2}},}
2914:) theorem states that a singular integral operator
2899:the operator given by multiplication by a function
1676:{\displaystyle |K(x,y)|\leq {\frac {C}{|x-y|^{n}}}}
483:{\displaystyle K_{i}(x)={\frac {x_{i}}{|x|^{n+1}}}}
3242:(2), The Johns Hopkins University Press: 289–309,
3075:
3023:
2985:
2843:
2548:
2422:
2296:singular integral operator of non-convolution type
2273:
1974:
1675:
1527:
1446:
1370:
1250:
1106:
966:Property 1. is needed to ensure that convolution (
963:) and satisfies a weak-type (1, 1) estimate.
938:
793:
700:
520:
482:
379:
121:
3154:
218:The archetypal singular integral operator is the
3558:
3416:
2298:associated to the Calderón–Zygmund kernel
2208:
1909:
1280:
1025:
823:
611:
290:
225:. It is given by convolution against the kernel
3490:, Princeton Mathematical Series, vol. 30,
3224:
3168:
554:Singular integral operators of convolution type
160:. Specifically, the singularity is such that |
2456:associated to a Calderón–Zygmund kernel
2761:
2728:
2697: > 0. An operator is said to be
2447:
2266:
2213:
2146:
2092:
1967:
1914:
1847:
1793:
544:and satisfy weak-type (1, 1) estimates.
3545:Notices of the American Mathematical Society
3286:
3106:Singular integral operators on closed curves
2527:
2520:
2498:
2482:
2452:A singular integral of non-convolution type
1438:
1432:
122:{\displaystyle T(f)(x)=\int K(x,y)f(y)\,dy,}
1553:
3135:
3133:
3131:
2287:Singular integrals of non-convolution type
1542:Singular integrals of non-convolution type
3191:
2810:
2625:)| ≤ 1, for all multi-indices |
2410:
2403:
2348:
1352:
1197:
1097:
1008:
1007:
920:
688:
367:
109:
3139:
540:. All of these operators are bounded on
207:
3360:
3328:
3128:
14:
3559:
2559:for all smooth compactly supported ƒ.
1381:then it can be shown that 1. follows.
548:Singular integrals of convolution type
3534:
3480:
3144:, New Jersey: Pearson Education, Inc.
3142:Classical and Modern Fourier Analysis
3123:
562:defined by convolution with a kernel
3117:
576:
2986:{\displaystyle M_{b_{2}}TM_{b_{1}}}
2573:
724:Suppose that the kernel satisfies:
24:
3091:is the transpose operator of
1467:
1216:
1000:
994:
768:
25:
3588:
3528:
3232:(1956), "On singular integrals",
1429:
3535:Stein, Elias M. (October 1998).
2448:Calderón–Zygmund operators
1419:
778:
3420:; Prössdorf, Siegfried (1986),
3235:American Journal of Mathematics
2870: > 0 such that Re(
2462:Calderón–Zygmund operator
3148:
3140:Grafakos, Loukas (2004), "7",
3067:
3054:
3018:
3005:
2807:
2801:
2798:
2785:
2772:
2766:
2756:
2743:
2494:
2488:
2400:
2394:
2388:
2376:
2370:
2364:
2345:
2339:
2336:
2330:
2324:
2318:
2260:
2246:
2238:
2219:
2191:
2172:
2139:
2120:
2112:
2098:
2078:
2058:
2044:
2040:
2023:
2014:
2002:
1995:
1961:
1942:
1934:
1920:
1892:
1873:
1840:
1821:
1813:
1799:
1779:
1759:
1745:
1741:
1724:
1715:
1703:
1696:
1660:
1645:
1631:
1627:
1615:
1608:
1506:
1497:
1483:
1479:
1473:
1463:
1441:
1414:
1348:
1344:
1338:
1331:
1315:
1307:
1194:
1188:
1166:
1158:
1094:
1088:
1082:
1076:
1061:
1053:
1032:
1018:
1012:
916:
912:
906:
897:
885:
878:
871:
863:
852:
844:
788:
773:
754:
685:
679:
673:
661:
646:
632:
618:
604:
598:
595:
589:
461:
452:
432:
426:
364:
358:
325:
311:
297:
273:
267:
264:
258:
106:
100:
94:
82:
70:
64:
61:
55:
13:
1:
3334:"Singular integral equations"
3162:
3076:{\displaystyle T^{t}(b_{2}),}
2472: > 0 such that
7:
3423:Singular Integral Operators
3099:
2862:. A function is said to be
968:
714:
10:
3593:
3496:Princeton University Press
2633: + 2. Denote by
951:Then it can be shown that
551:
211:
2854:for all normalised bumps
3111:
3024:{\displaystyle T(b_{1})}
1554:Calderón–Zygmund kernels
982:principal value integral
574:\{0}, in the sense that
2866:if there is a constant
2701:if there is a constant
18:Calderón–Zygmund kernel
3077:
3025:
2987:
2938:
2845:
2566:with 1 <
2550:
2468:, that is, there is a
2464:when it is bounded on
2424:
2275:
1976:
1677:
1597:
1591: > 0 and
1529:
1448:
1372:
1252:
1108:
940:
795:
702:
522:
484:
381:
132:whose kernel function
123:
3469:, (European edition:
3078:
3026:
2988:
2846:
2605:is a smooth function
2551:
2425:
2276:
1977:
1678:
1530:
1449:
1373:
1253:
1109:
974:tempered distribution
941:
796:
703:
523:
521:{\displaystyle x_{i}}
485:
382:
208:The Hilbert transform
180:| asymptotically as |
124:
3498:, pp. XIV+287,
3396:, pp. XII+255,
3041:
2999:
2943:
2712:
2479:
2309:
2168: whenever
1991:
1869: whenever
1692:
1604:
1459:
1395:
1276:
1136:
990:
819:
809:condition: for some
745:
583:
505:
413:
252:
49:
3418:Mikhlin, Solomon G.
3362:Mikhlin, Solomon G.
3330:Mikhlin, Solomon G.
2570: < ∞.
1595: > 0.
813: > 0,
192: −
184: −
176: −
152:along the diagonal
3567:Singular integrals
3193:10.1007/BF02392130
3073:
3021:
2993:is weakly bounded;
2983:
2841:
2546:
2420:
2271:
1972:
1673:
1525:
1444:
1368:
1294:
1248:
1119:Fourier multiplier
1117:is a well-defined
1104:
1046:
936:
837:
791:
698:
625:
568:locally integrable
518:
480:
377:
304:
245:. More precisely,
233:) = 1/(π
119:
33:singular integrals
3572:Harmonic analysis
2206:
2169:
2164:
1907:
1870:
1865:
1671:
1523:
1279:
1215:
1024:
822:
757:
734:Fourier transform
732:condition on the
722:
721:
610:
532:-th component of
478:
353:
289:
287:
220:Hilbert transform
214:Hilbert transform
41:integral operator
37:harmonic analysis
16:(Redirected from
3584:
3553:
3541:
3524:
3468:
3412:
3352:
3324:
3282:
3220:
3195:
3179:Acta Mathematica
3157:
3156:
3152:
3146:
3145:
3137:
3126:
3125:
3121:
3082:
3080:
3079:
3074:
3066:
3065:
3053:
3052:
3030:
3028:
3027:
3022:
3017:
3016:
2992:
2990:
2989:
2984:
2982:
2981:
2980:
2979:
2962:
2961:
2960:
2959:
2850:
2848:
2847:
2842:
2840:
2839:
2821:
2817:
2797:
2796:
2784:
2783:
2765:
2764:
2755:
2754:
2742:
2741:
2732:
2731:
2555:
2553:
2552:
2547:
2542:
2541:
2540:
2539:
2513:
2512:
2511:
2510:
2429:
2427:
2426:
2421:
2294:is said to be a
2280:
2278:
2277:
2272:
2270:
2269:
2263:
2249:
2241:
2236:
2222:
2217:
2216:
2207:
2199:
2194:
2189:
2175:
2170:
2167:
2165:
2163:
2162:
2161:
2150:
2149:
2142:
2137:
2123:
2115:
2101:
2096:
2095:
2088:
2087:
2086:
2081:
2075:
2061:
2052:
2047:
2039:
1998:
1981:
1979:
1978:
1973:
1971:
1970:
1964:
1953:
1945:
1937:
1923:
1918:
1917:
1908:
1900:
1895:
1890:
1876:
1871:
1868:
1866:
1864:
1863:
1862:
1851:
1850:
1843:
1832:
1824:
1816:
1802:
1797:
1796:
1789:
1788:
1787:
1782:
1776:
1762:
1753:
1748:
1734:
1699:
1682:
1680:
1679:
1674:
1672:
1670:
1669:
1668:
1663:
1648:
1639:
1634:
1611:
1576:is said to be a
1575:
1534:
1532:
1531:
1526:
1524:
1522:
1521:
1520:
1509:
1500:
1491:
1486:
1466:
1453:
1451:
1450:
1445:
1428:
1427:
1422:
1413:
1412:
1377:
1375:
1374:
1369:
1351:
1334:
1329:
1328:
1318:
1310:
1293:
1257:
1255:
1254:
1249:
1241:
1240:
1228:
1227:
1213:
1184:
1183:
1182:
1181:
1169:
1161:
1153:
1152:
1113:
1111:
1110:
1105:
1072:
1071:
1064:
1056:
1045:
1044:
1043:
1006:
945:
943:
942:
937:
919:
881:
876:
875:
874:
866:
855:
847:
836:
800:
798:
797:
792:
787:
786:
781:
772:
771:
759:
758:
750:
716:
707:
705:
704:
699:
657:
656:
649:
635:
624:
577:
527:
525:
524:
519:
517:
516:
489:
487:
486:
481:
479:
477:
476:
475:
464:
455:
449:
448:
439:
425:
424:
402:) = 1/
394:, which replace
392:Riesz transforms
386:
384:
383:
378:
354:
352:
338:
336:
335:
328:
314:
303:
288:
280:
128:
126:
125:
120:
21:
3592:
3591:
3587:
3586:
3585:
3583:
3582:
3581:
3557:
3556:
3552:(9): 1130–1140.
3539:
3531:
3506:
3450:
3442:, p. 528,
3440:Springer Verlag
3306:
3288:Coifman, Ronald
3248:10.2307/2372517
3226:Calderon, A. P.
3170:Calderon, A. P.
3165:
3160:
3153:
3149:
3138:
3129:
3122:
3118:
3114:
3102:
3061:
3057:
3048:
3044:
3042:
3039:
3038:
3012:
3008:
3000:
2997:
2996:
2975:
2971:
2970:
2966:
2955:
2951:
2950:
2946:
2944:
2941:
2940:
2935:
2928:
2898:
2832:
2828:
2792:
2788:
2779:
2775:
2760:
2759:
2750:
2746:
2737:
2733:
2727:
2726:
2719:
2715:
2713:
2710:
2709:
2665:
2603:normalised bump
2584:
2535:
2531:
2530:
2526:
2506:
2502:
2501:
2497:
2480:
2477:
2476:
2450:
2310:
2307:
2306:
2289:
2265:
2264:
2259:
2245:
2237:
2229:
2218:
2212:
2211:
2198:
2190:
2182:
2171:
2166:
2151:
2145:
2144:
2143:
2138:
2130:
2119:
2111:
2097:
2091:
2090:
2089:
2082:
2077:
2076:
2068:
2057:
2053:
2051:
2043:
2032:
1994:
1992:
1989:
1988:
1966:
1965:
1960:
1946:
1941:
1933:
1919:
1913:
1912:
1899:
1891:
1883:
1872:
1867:
1852:
1846:
1845:
1844:
1839:
1825:
1820:
1812:
1798:
1792:
1791:
1790:
1783:
1778:
1777:
1769:
1758:
1754:
1752:
1744:
1727:
1695:
1693:
1690:
1689:
1664:
1659:
1658:
1644:
1643:
1638:
1630:
1607:
1605:
1602:
1601:
1559:
1556:
1544:
1510:
1505:
1504:
1496:
1495:
1490:
1482:
1462:
1460:
1457:
1456:
1423:
1418:
1417:
1408:
1404:
1396:
1393:
1392:
1347:
1330:
1314:
1306:
1299:
1295:
1283:
1277:
1274:
1273:
1236:
1232:
1223:
1219:
1177:
1173:
1165:
1157:
1148:
1144:
1143:
1139:
1137:
1134:
1133:
1060:
1052:
1051:
1047:
1039:
1035:
1028:
993:
991:
988:
987:
915:
877:
870:
862:
851:
843:
842:
838:
826:
820:
817:
816:
782:
777:
776:
767:
763:
749:
748:
746:
743:
742:
645:
631:
630:
626:
614:
584:
581:
580:
556:
550:
512:
508:
506:
503:
502:
465:
460:
459:
451:
450:
444:
440:
438:
420:
416:
414:
411:
410:
342:
337:
324:
310:
309:
305:
293:
279:
253:
250:
249:
216:
210:
172:)| is of size |
50:
47:
46:
35:are central to
23:
22:
15:
12:
11:
5:
3590:
3580:
3579:
3574:
3569:
3555:
3554:
3530:
3529:External links
3527:
3526:
3525:
3504:
3478:
3448:
3414:
3394:Pergamon Press
3358:
3344:(25): 29–112,
3326:
3304:
3284:
3222:
3164:
3161:
3159:
3158:
3147:
3127:
3115:
3113:
3110:
3109:
3108:
3101:
3098:
3097:
3096:
3072:
3069:
3064:
3060:
3056:
3051:
3047:
3036:
3020:
3015:
3011:
3007:
3004:
2994:
2978:
2974:
2969:
2965:
2958:
2954:
2949:
2933:
2926:
2894:
2878:) ≥
2852:
2851:
2838:
2835:
2831:
2827:
2824:
2820:
2816:
2813:
2809:
2806:
2803:
2800:
2795:
2791:
2787:
2782:
2778:
2774:
2771:
2768:
2763:
2758:
2753:
2749:
2745:
2740:
2736:
2730:
2725:
2722:
2718:
2699:weakly bounded
2670:) =
2661:
2645:) =
2629:| ≤
2583:
2572:
2557:
2556:
2545:
2538:
2534:
2529:
2525:
2522:
2519:
2516:
2509:
2505:
2500:
2496:
2493:
2490:
2487:
2484:
2449:
2446:
2431:
2430:
2419:
2416:
2413:
2409:
2406:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2288:
2285:
2284:
2283:
2282:
2281:
2268:
2262:
2258:
2255:
2252:
2248:
2244:
2240:
2235:
2232:
2228:
2225:
2221:
2215:
2210:
2205:
2202:
2197:
2193:
2188:
2185:
2181:
2178:
2174:
2160:
2157:
2154:
2148:
2141:
2136:
2133:
2129:
2126:
2122:
2118:
2114:
2110:
2107:
2104:
2100:
2094:
2085:
2080:
2074:
2071:
2067:
2064:
2060:
2056:
2050:
2046:
2042:
2038:
2035:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1997:
1984:
1983:
1982:
1969:
1963:
1959:
1956:
1952:
1949:
1944:
1940:
1936:
1932:
1929:
1926:
1922:
1916:
1911:
1906:
1903:
1898:
1894:
1889:
1886:
1882:
1879:
1875:
1861:
1858:
1855:
1849:
1842:
1838:
1835:
1831:
1828:
1823:
1819:
1815:
1811:
1808:
1805:
1801:
1795:
1786:
1781:
1775:
1772:
1768:
1765:
1761:
1757:
1751:
1747:
1743:
1740:
1737:
1733:
1730:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1698:
1685:
1684:
1683:
1667:
1662:
1657:
1654:
1651:
1647:
1642:
1637:
1633:
1629:
1626:
1623:
1620:
1617:
1614:
1610:
1555:
1552:
1543:
1540:
1536:
1535:
1519:
1516:
1513:
1508:
1503:
1499:
1494:
1489:
1485:
1481:
1478:
1475:
1472:
1469:
1465:
1454:
1443:
1440:
1437:
1434:
1431:
1426:
1421:
1416:
1411:
1407:
1403:
1400:
1379:
1378:
1367:
1364:
1361:
1358:
1355:
1350:
1346:
1343:
1340:
1337:
1333:
1327:
1324:
1321:
1317:
1313:
1309:
1305:
1302:
1298:
1292:
1289:
1286:
1282:
1259:
1258:
1247:
1244:
1239:
1235:
1231:
1226:
1222:
1218:
1212:
1209:
1206:
1203:
1200:
1196:
1193:
1190:
1187:
1180:
1176:
1172:
1168:
1164:
1160:
1156:
1151:
1147:
1142:
1115:
1114:
1103:
1100:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1070:
1067:
1063:
1059:
1055:
1050:
1042:
1038:
1034:
1031:
1027:
1023:
1020:
1017:
1014:
1011:
1005:
1002:
999:
996:
955:is bounded on
949:
948:
947:
946:
935:
932:
929:
926:
923:
918:
914:
911:
908:
905:
902:
899:
896:
893:
890:
887:
884:
880:
873:
869:
865:
861:
858:
854:
850:
846:
841:
835:
832:
829:
825:
803:
802:
801:
790:
785:
780:
775:
770:
766:
762:
756:
753:
720:
719:
710:
708:
697:
694:
691:
687:
684:
681:
678:
675:
672:
669:
666:
663:
660:
655:
652:
648:
644:
641:
638:
634:
629:
623:
620:
617:
613:
609:
606:
603:
600:
597:
594:
591:
588:
552:Main article:
549:
546:
515:
511:
491:
490:
474:
471:
468:
463:
458:
454:
447:
443:
437:
434:
431:
428:
423:
419:
388:
387:
376:
373:
370:
366:
363:
360:
357:
351:
348:
345:
341:
334:
331:
327:
323:
320:
317:
313:
308:
302:
299:
296:
292:
286:
283:
278:
275:
272:
269:
266:
263:
260:
257:
212:Main article:
209:
206:
130:
129:
118:
115:
112:
108:
105:
102:
99:
96:
93:
90:
87:
84:
81:
78:
75:
72:
69:
66:
63:
60:
57:
54:
9:
6:
4:
3:
2:
3589:
3578:
3577:Real analysis
3575:
3573:
3570:
3568:
3565:
3564:
3562:
3551:
3547:
3546:
3538:
3533:
3532:
3523:
3519:
3515:
3511:
3507:
3505:0-691-08079-8
3501:
3497:
3493:
3492:Princeton, NJ
3489:
3488:
3483:
3479:
3476:
3475:3-540-15967-3
3472:
3467:
3463:
3459:
3455:
3451:
3449:0-387-15967-3
3445:
3441:
3437:
3436:New York City
3433:
3429:
3425:
3424:
3419:
3415:
3411:
3407:
3403:
3399:
3395:
3391:
3387:
3383:
3382:New York City
3379:
3375:
3371:
3367:
3363:
3359:
3356:
3351:
3347:
3343:
3339:
3335:
3331:
3327:
3323:
3319:
3315:
3311:
3307:
3305:0-521-42001-6
3301:
3297:
3293:
3289:
3285:
3281:
3277:
3273:
3269:
3265:
3261:
3257:
3253:
3249:
3245:
3241:
3237:
3236:
3231:
3227:
3223:
3219:
3215:
3211:
3207:
3203:
3199:
3194:
3189:
3186:(1): 85–139,
3185:
3181:
3180:
3175:
3171:
3167:
3166:
3151:
3143:
3136:
3134:
3132:
3120:
3116:
3107:
3104:
3103:
3094:
3090:
3086:
3070:
3062:
3058:
3049:
3045:
3037:
3034:
3013:
3009:
3002:
2995:
2976:
2972:
2967:
2963:
2956:
2952:
2947:
2939:
2937:
2932:
2925:
2921:
2917:
2913:
2909:
2904:
2902:
2897:
2893:
2889:
2885:
2881:
2877:
2873:
2869:
2865:
2861:
2857:
2836:
2833:
2829:
2825:
2822:
2818:
2814:
2811:
2804:
2793:
2789:
2780:
2776:
2769:
2751:
2747:
2738:
2734:
2723:
2720:
2716:
2708:
2707:
2706:
2704:
2700:
2696:
2692:
2688:
2684:
2680:
2676:
2673:
2669:
2664:
2660:
2656:
2653: −
2652:
2648:
2644:
2640:
2636:
2632:
2628:
2624:
2620:
2616:
2612:
2608:
2604:
2599:
2597:
2593:
2589:
2581:
2577:
2571:
2569:
2565:
2560:
2543:
2536:
2532:
2523:
2517:
2514:
2507:
2503:
2491:
2485:
2475:
2474:
2473:
2471:
2467:
2463:
2459:
2455:
2445:
2444:
2440:
2436:
2417:
2414:
2411:
2407:
2404:
2397:
2391:
2385:
2382:
2379:
2373:
2367:
2361:
2358:
2355:
2352:
2349:
2342:
2333:
2327:
2321:
2315:
2312:
2305:
2304:
2303:
2301:
2297:
2293:
2256:
2253:
2250:
2242:
2233:
2230:
2226:
2223:
2203:
2200:
2195:
2186:
2183:
2179:
2176:
2158:
2155:
2152:
2134:
2131:
2127:
2124:
2116:
2108:
2105:
2102:
2083:
2072:
2069:
2065:
2062:
2054:
2048:
2036:
2033:
2029:
2026:
2020:
2017:
2011:
2008:
2005:
1999:
1987:
1986:
1985:
1957:
1954:
1950:
1947:
1938:
1930:
1927:
1924:
1904:
1901:
1896:
1887:
1884:
1880:
1877:
1859:
1856:
1853:
1836:
1833:
1829:
1826:
1817:
1809:
1806:
1803:
1784:
1773:
1770:
1766:
1763:
1755:
1749:
1738:
1735:
1731:
1728:
1721:
1718:
1712:
1709:
1706:
1700:
1688:
1687:
1686:
1665:
1655:
1652:
1649:
1640:
1635:
1624:
1621:
1618:
1612:
1600:
1599:
1598:
1596:
1594:
1590:
1586:
1584:
1580:
1574:
1570:
1566:
1562:
1551:
1549:
1539:
1517:
1514:
1511:
1501:
1492:
1487:
1476:
1470:
1455:
1435:
1424:
1409:
1405:
1401:
1398:
1391:
1390:
1389:
1388:can be used:
1387:
1382:
1365:
1362:
1359:
1356:
1353:
1341:
1335:
1325:
1322:
1319:
1311:
1303:
1300:
1296:
1290:
1287:
1284:
1272:
1271:
1270:
1268:
1264:
1245:
1242:
1237:
1233:
1229:
1224:
1220:
1210:
1207:
1204:
1201:
1198:
1191:
1185:
1178:
1174:
1170:
1162:
1154:
1149:
1145:
1140:
1132:
1131:
1130:
1128:
1124:
1120:
1101:
1098:
1091:
1085:
1079:
1073:
1068:
1065:
1057:
1048:
1040:
1036:
1029:
1021:
1015:
1009:
1003:
997:
986:
985:
984:
983:
980:given by the
979:
975:
971:
970:
964:
962:
958:
954:
933:
930:
927:
924:
921:
909:
903:
900:
894:
891:
888:
882:
867:
859:
856:
848:
839:
833:
830:
827:
815:
814:
812:
808:
804:
783:
764:
760:
751:
741:
740:
739:
735:
731:
727:
726:
725:
718:
711:
709:
695:
692:
689:
682:
676:
670:
667:
664:
658:
653:
650:
642:
639:
636:
627:
621:
615:
607:
601:
592:
586:
579:
578:
575:
573:
569:
565:
561:
555:
545:
543:
539:
535:
531:
513:
509:
500:
496:
472:
469:
466:
456:
445:
441:
435:
429:
421:
417:
409:
408:
407:
405:
401:
397:
393:
374:
371:
368:
361:
355:
349:
346:
343:
339:
332:
329:
321:
318:
315:
306:
300:
294:
284:
281:
276:
270:
261:
255:
248:
247:
246:
244:
240:
236:
232:
228:
224:
221:
215:
205:
203:
199:
195:
191:
187:
183:
179:
175:
171:
167:
163:
159:
156: =
155:
151:
147:
144: →
143:
139:
135:
116:
113:
110:
103:
97:
91:
88:
85:
79:
76:
73:
67:
58:
52:
45:
44:
43:
42:
38:
34:
30:
19:
3549:
3543:
3486:
3482:Stein, Elias
3422:
3365:
3341:
3337:
3295:
3239:
3233:
3183:
3177:
3150:
3141:
3119:
3092:
3088:
2930:
2923:
2919:
2915:
2911:
2907:
2905:
2900:
2895:
2891:
2890:. Denote by
2887:
2883:
2879:
2875:
2871:
2867:
2863:
2859:
2855:
2853:
2702:
2698:
2694:
2690:
2686:
2682:
2678:
2674:
2671:
2667:
2662:
2658:
2654:
2650:
2646:
2642:
2638:
2634:
2630:
2626:
2622:
2618:
2614:
2610:
2606:
2602:
2600:
2595:
2591:
2587:
2585:
2579:
2575:
2567:
2563:
2561:
2558:
2469:
2465:
2461:
2460:is called a
2457:
2453:
2451:
2442:
2438:
2434:
2432:
2299:
2295:
2291:
2290:
1592:
1588:
1577:
1572:
1568:
1564:
1560:
1557:
1547:
1545:
1537:
1385:
1383:
1380:
1267:odd function
1262:
1260:
1127:cancellation
1126:
1122:
1116:
977:
967:
965:
960:
956:
952:
950:
810:
806:
737:
729:
723:
712:
571:
563:
559:
557:
541:
537:
533:
529:
498:
494:
492:
403:
399:
395:
389:
242:
238:
234:
230:
226:
222:
217:
201:
197:
193:
189:
185:
181:
177:
173:
169:
165:
161:
157:
153:
145:
141:
137:
133:
131:
32:
26:
3292:Meyer, Yves
3230:Zygmund, A.
3174:Zygmund, A.
1558:A function
972:) with the
29:mathematics
3561:Categories
3522:0207.13501
3466:0612.47024
3432:Heidelberg
3410:0129.07701
3322:0916.42023
3280:0072.11501
3218:0047.10201
3163:References
2705:such that
2685:) for all
1129:condition
976:p.v.
807:smoothness
497:= 1, ...,
3390:Frankfurt
3378:Edinburgh
3256:0002-9327
3202:0001-5962
2864:accretive
2834:−
2823:≤
2790:ψ
2777:τ
2748:φ
2735:τ
2721:∫
2582:) theorem
2528:‖
2521:‖
2515:≤
2499:‖
2483:‖
2433:whenever
2359:∬
2313:∫
2254:−
2227:−
2196:≤
2180:−
2159:δ
2128:−
2106:−
2084:δ
2066:−
2049:≤
2018:−
1955:−
1928:−
1897:≤
1881:−
1860:δ
1834:−
1807:−
1785:δ
1767:−
1750:≤
1719:−
1653:−
1636:≤
1488:≤
1468:∇
1430:∖
1402:∈
1360:≤
1297:∫
1217:∀
1141:∫
1074:ϕ
1069:ϵ
1049:∫
1033:→
1030:ϵ
1016:ϕ
928:≤
901:−
892:−
840:∫
831:≠
769:∞
761:∈
755:^
668:−
654:ε
640:−
628:∫
619:→
616:ε
347:−
333:ε
319:−
307:∫
298:→
295:ε
285:π
77:∫
3484:(1970),
3364:(1965),
3332:(1948),
3294:(1997),
3100:See also
3087:, where
2882:for all
2234:′
2187:′
2135:′
2073:′
2037:′
1951:′
1888:′
1830:′
1774:′
1732:′
1579:Calderón
1563: :
566:that is
150:singular
136: :
3514:0290095
3458:0867687
3402:0185399
3355:Russian
3350:0027429
3314:1456993
3272:0084633
3264:2372517
3210:0052553
1583:Zygmund
528:is the
168:,
3520:
3512:
3502:
3473:
3464:
3456:
3446:
3428:Berlin
3408:
3400:
3374:London
3370:Oxford
3348:
3320:
3312:
3302:
3278:
3270:
3262:
3254:
3216:
3208:
3200:
3083:is in
3031:is in
2657:) and
2617:
1585:kernel
1265:is an
1214:
493:where
237:) for
140:×
3540:(PDF)
3386:Paris
3260:JSTOR
3112:Notes
501:and
406:with
3500:ISBN
3471:ISBN
3444:ISBN
3353:(in
3300:ISBN
3252:ISSN
3198:ISSN
2929:and
2906:The
2858:and
2693:and
2586:The
2574:The
2437:and
1320:<
1304:<
1288:>
1243:>
1171:<
1155:<
1066:>
857:>
805:The
730:size
728:The
651:>
330:>
3518:Zbl
3462:Zbl
3406:Zbl
3338:UMN
3318:Zbl
3276:Zbl
3244:doi
3214:Zbl
3188:doi
3085:BMO
3033:BMO
2886:in
2689:in
2609:on
2302:if
2209:max
1910:max
1281:sup
1121:on
1026:lim
824:sup
736:of
612:lim
570:on
536:in
291:lim
241:in
204:).
148:is
27:In
3563::
3550:45
3548:.
3542:.
3516:,
3510:MR
3508:,
3494::
3477:).
3460:,
3454:MR
3452:,
3438::
3426:,
3404:,
3398:MR
3392::
3357:).
3346:MR
3340:,
3336:,
3316:,
3310:MR
3308:,
3290:;
3274:,
3268:MR
3266:,
3258:,
3250:,
3240:78
3238:,
3228:;
3212:,
3206:MR
3204:,
3196:,
3184:88
3182:,
3172:;
3130:^
2936::
2903:.
2874:)(
2641:)(
2601:A
1571:→
1550:.
31:,
3434:–
3430:–
3413:.
3388:–
3384:–
3380:–
3376:–
3372:–
3342:3
3325:.
3283:.
3246::
3221:.
3190::
3095:.
3093:T
3089:T
3071:,
3068:)
3063:2
3059:b
3055:(
3050:t
3046:T
3035:;
3019:)
3014:1
3010:b
3006:(
3003:T
2977:1
2973:b
2968:M
2964:T
2957:2
2953:b
2948:M
2934:2
2931:b
2927:1
2924:b
2920:L
2916:T
2912:b
2910:(
2908:T
2901:b
2896:b
2892:M
2888:R
2884:x
2880:c
2876:x
2872:b
2868:c
2860:ψ
2856:φ
2837:n
2830:r
2826:C
2819:|
2815:y
2812:d
2808:)
2805:y
2802:(
2799:)
2794:r
2786:(
2781:x
2773:)
2770:y
2767:(
2762:)
2757:)
2752:r
2744:(
2739:x
2729:(
2724:T
2717:|
2703:C
2695:r
2691:R
2687:x
2683:r
2681:/
2679:x
2677:(
2675:φ
2672:r
2668:x
2666:(
2663:r
2659:φ
2655:x
2651:y
2649:(
2647:φ
2643:y
2639:φ
2637:(
2635:τ
2631:n
2627:α
2623:x
2621:(
2619:φ
2615:∂
2611:R
2607:φ
2596:L
2592:b
2590:(
2588:T
2580:b
2578:(
2576:T
2568:p
2564:L
2544:,
2537:2
2533:L
2524:f
2518:C
2508:2
2504:L
2495:)
2492:f
2489:(
2486:T
2470:C
2466:L
2458:K
2454:T
2443:L
2439:g
2435:f
2418:,
2415:x
2412:d
2408:y
2405:d
2401:)
2398:y
2395:(
2392:f
2389:)
2386:y
2383:,
2380:x
2377:(
2374:K
2371:)
2368:x
2365:(
2362:g
2356:=
2353:x
2350:d
2346:)
2343:x
2340:(
2337:)
2334:f
2331:(
2328:T
2325:)
2322:x
2319:(
2316:g
2300:K
2292:T
2267:)
2261:|
2257:y
2251:x
2247:|
2243:,
2239:|
2231:y
2224:x
2220:|
2214:(
2204:2
2201:1
2192:|
2184:y
2177:y
2173:|
2156:+
2153:n
2147:)
2140:|
2132:y
2125:x
2121:|
2117:+
2113:|
2109:y
2103:x
2099:|
2093:(
2079:|
2070:y
2063:y
2059:|
2055:C
2045:|
2041:)
2034:y
2030:,
2027:x
2024:(
2021:K
2015:)
2012:y
2009:,
2006:x
2003:(
2000:K
1996:|
1968:)
1962:|
1958:y
1948:x
1943:|
1939:,
1935:|
1931:y
1925:x
1921:|
1915:(
1905:2
1902:1
1893:|
1885:x
1878:x
1874:|
1857:+
1854:n
1848:)
1841:|
1837:y
1827:x
1822:|
1818:+
1814:|
1810:y
1804:x
1800:|
1794:(
1780:|
1771:x
1764:x
1760:|
1756:C
1746:|
1742:)
1739:y
1736:,
1729:x
1725:(
1722:K
1716:)
1713:y
1710:,
1707:x
1704:(
1701:K
1697:|
1666:n
1661:|
1656:y
1650:x
1646:|
1641:C
1632:|
1628:)
1625:y
1622:,
1619:x
1616:(
1613:K
1609:|
1593:δ
1589:C
1581:–
1573:R
1569:R
1567:×
1565:R
1561:K
1548:L
1518:1
1515:+
1512:n
1507:|
1502:x
1498:|
1493:C
1484:|
1480:)
1477:x
1474:(
1471:K
1464:|
1442:)
1439:}
1436:0
1433:{
1425:n
1420:R
1415:(
1410:1
1406:C
1399:K
1386:K
1366:,
1363:C
1357:x
1354:d
1349:|
1345:)
1342:x
1339:(
1336:K
1332:|
1326:R
1323:2
1316:|
1312:x
1308:|
1301:R
1291:0
1285:R
1263:K
1246:0
1238:2
1234:R
1230:,
1225:1
1221:R
1211:,
1208:0
1205:=
1202:x
1199:d
1195:)
1192:x
1189:(
1186:K
1179:2
1175:R
1167:|
1163:x
1159:|
1150:1
1146:R
1123:L
1102:x
1099:d
1095:)
1092:x
1089:(
1086:K
1083:)
1080:x
1077:(
1062:|
1058:x
1054:|
1041:+
1037:0
1022:=
1019:]
1013:[
1010:K
1004:.
1001:v
998:.
995:p
978:K
969:1
961:R
959:(
957:L
953:T
934:.
931:C
925:x
922:d
917:|
913:)
910:x
907:(
904:K
898:)
895:y
889:x
886:(
883:K
879:|
872:|
868:y
864:|
860:2
853:|
849:x
845:|
834:0
828:y
811:C
789:)
784:n
779:R
774:(
765:L
752:K
738:K
717:)
715:1
713:(
696:.
693:y
690:d
686:)
683:y
680:(
677:f
674:)
671:y
665:x
662:(
659:K
647:|
643:x
637:y
633:|
622:0
608:=
605:)
602:x
599:(
596:)
593:f
590:(
587:T
572:R
564:K
560:T
542:L
538:R
534:x
530:i
514:i
510:x
499:n
495:i
473:1
470:+
467:n
462:|
457:x
453:|
446:i
442:x
436:=
433:)
430:x
427:(
422:i
418:K
404:x
400:x
398:(
396:K
375:.
372:y
369:d
365:)
362:y
359:(
356:f
350:y
344:x
340:1
326:|
322:y
316:x
312:|
301:0
282:1
277:=
274:)
271:x
268:(
265:)
262:f
259:(
256:H
243:R
239:x
235:x
231:x
229:(
227:K
223:H
202:R
200:(
198:L
194:x
190:y
186:y
182:x
178:y
174:x
170:y
166:x
164:(
162:K
158:y
154:x
146:R
142:R
138:R
134:K
117:,
114:y
111:d
107:)
104:y
101:(
98:f
95:)
92:y
89:,
86:x
83:(
80:K
74:=
71:)
68:x
65:(
62:)
59:f
56:(
53:T
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.