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W2155111099 abstract "Some multilinear maximal functions and the generalized Calderón-Zygmund operators and their commutators with nonsmooth kernels are studied. The purpose of this paper is to establish that these operators are bounded on certain product Morrey spaces <svg style=vertical-align:-2.3205pt;width:57.162498px; id=M1 height=19.65 version=1.1 viewBox=0 0 57.162498 19.65 width=57.162498 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,16.7)><path id=x1D43F d=M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z /></g> <g transform=matrix(.012,-0,0,-.012,9.763,8.537)><path id=x1D45D d=M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36
t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z /></g><g transform=matrix(.012,-0,0,-.012,16.818,8.537)><path id=x2C d=M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z /></g><g transform=matrix(.012,-0,0,-.012,19.531,8.537)><path id=x1D458 d=M480 416q0 -21 -18 -41q-9 -11 -17 -7q-20 9 -42 9q-62 0 -140 -78q23 -69 88 -192q17 -31 27 -42t20 -11q16 0 62 46l17 -20q-64 -92 -119 -92q-35 0 -70 66q-41 73 -84 187q-36 -30 -62 -61q-27 -115 -35 -172q-41 -8 -78 -20l-6 6l140 612q7 28 0.5 34t-37.5 7l-34 1
l5 26q38 4 74 13.5t57 17t25 7.5q12 0 4 -32l-104 -443h2q35 38 97 93q39 35 65.5 56t62 41.5t58.5 20.5q19 0 30.5 -10t11.5 -22z /></g> <g transform=matrix(.017,-0,0,-.017,26.15,16.7)><path id=x28 d=M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z /></g><g transform=matrix(.017,-0,0,-.017,32.032,16.7)><path id=x211D d=M751 18q-1 -4 -2 -13t-2 -13q-24 0 -35 1q-57 3 -92.5 23t-69.5 68q-21 30 -101 160q-13 22 -30.5 31t-54.5 9h-32v-159q0 -62 14 -77t74 -20v-28h-382v28q62 5 76 19.5t14 77.5v400q0 63 -14 77.5t-76 19.5v28h372q109 0 159 -34q67 -42 67 -131q0 -115 -127 -172
q27 -52 89 -144q49 -71 81 -104q31 -36 72 -47zM543 469q0 76 -42 112t-104 36q-41 0 -54 -10q-11 -7 -11 -44v-247h48q73 0 108 29q55 42 55 124zM290 131v390q0 62 -7.5 79.5t-30.5 17.5q-24 0 -31.5 -17.5t-7.5 -79.5v-390q0 -62 7.5 -79.5t31.5 -17.5q23 0 30.5 17.5
t7.5 79.5z /></g> <g transform=matrix(.012,-0,0,-.012,44.625,8.537)><path id=x1D45B d=M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2
q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z /></g> <g transform=matrix(.017,-0,0,-.017,51.225,16.7)><path id=x29 d=M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z /></g> </svg>. Based on the boundedness of these operators from <svg style=vertical-align:-3.39066pt;width:163.25px; id=M2 height=18.674999 version=1.1 viewBox=0 0 163.25 18.674999 width=163.25 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,14.388)><use xlink:href=#x1D43F/></g> <g transform=matrix(.012,-0,0,-.012,9.763,6.225)><use xlink:href=#x1D45D/></g> <g transform=matrix(.008,-0,0,-.008,16.825,9.712)><path id=x31 d=M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z /></g> <g transform=matrix(.017,-0,0,-.017,22.15,14.388)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,28.032,14.388)><path id=x1D714 d=M615 290q0 -148 -114 -245q-65 -57 -125 -57q-39 0 -62 27t-29 70q-31 -41 -72 -69t-74 -28q-49 0 -82.5 43t-33.5 124q0 97 55 174.5t148 124.5l15 -24q-59 -45 -94 -113t-35 -146q0 -58 19.5 -88.5t48.5 -30.5q33 0 62 34.5t45 101.5q16 73 20 135l80 17l5 -6
q-11 -37 -23.5 -100t-12.5 -92q0 -47 16.5 -71.5t44.5 -24.5q62 0 95.5 67t33.5 160q0 107 -63 152l1 9q14 14 45 14q29 0 57.5 -44t28.5 -114z /></g> <g transform=matrix(.012,-0,0,-.012,38.875,18.463)><path id=x1D456 d=M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30
q16 0 76 48z /></g> <g transform=matrix(.017,-0,0,-.017,42.75,14.388)><use xlink:href=#x29/></g><g transform=matrix(.017,-0,0,-.017,52.405,14.388)><path id=xD7 d=M528 54l-36 -38l-198 201l-198 -201l-36 38l197 200l-197 201l36 38l198 -202l198 202l36 -38l-197 -201z /></g><g transform=matrix(.017,-0,0,-.017,68.979,14.388)><path id=x22EF d=M215 255q0 -26 -16 -42t-40 -16q-25 0 -40.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5zM541 255q0 -26 -16 -42t-40 -16q-25 0 -40.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5zM867 255q0 -26 -16 -42t-40 -16q-25 0 -40.5 16t-15.5 42
q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5z /></g><g transform=matrix(.017,-0,0,-.017,92.08,14.388)><use xlink:href=#xD7/></g><g transform=matrix(.017,-0,0,-.017,105.849,14.388)><use xlink:href=#x1D43F/></g> <g transform=matrix(.012,-0,0,-.012,115.562,6.225)><use xlink:href=#x1D45D/></g> <g transform=matrix(.008,-0,0,-.008,122.613,9.712)><path id=x1D45A d=M766 88q-40 -45 -83.5 -72.5t-63.5 -27.5q-39 0 -16 103l49 224q15 68 -10 68q-36 0 -112.5 -78.5t-105.5 -133.5q-27 -101 -38 -165q-35 -4 -77 -18l-6 6q42 153 70 288q12 55 10 78t-19 23q-38 0 -114 -80.5t-101 -131.5q-24 -68 -41 -165q-36 -3 -76 -18l-8 6
q56 202 87 347q7 33 -3 33q-8 0 -31.5 -17.5t-41.5 -35.5l-12 28q42 45 84 72t63 27q44 0 10 -124l-20 -75h2q92 121 193 178q36 21 64 21q62 0 28 -159l-8 -37h2q50 67 103.5 112t94.5 65q39 19 62 19q60 0 23 -156l-45 -189q-9 -39 2 -39q17 0 71 49z /></g> <g transform=matrix(.017,-0,0,-.017,130.562,14.388)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,136.444,14.388)><use xlink:href=#x1D714/></g> <g transform=matrix(.012,-0,0,-.012,147.288,18.463)><use xlink:href=#x1D45A/></g> <g transform=matrix(.017,-0,0,-.017,157.3,14.388)><use xlink:href=#x29/></g> </svg> to <svg style=vertical-align:-6.57922pt;width:100.525px; id=M3 height=28.487499 version=1.1 viewBox=0 0 100.525 28.487499 width=100.525 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,20.237)><use xlink:href=#x1D43F/></g> <g transform=matrix(.012,-0,0,-.012,9.763,12.075)><use xlink:href=#x1D45D/></g> <g transform=matrix(.017,-0,0,-.017,17.45,20.237)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,23.332,20.237)><path id=x220F d=M851 711q-62 -5 -76.5 -21.5t-14.5 -78.5v-710q0 -63 14.5 -80t73.5 -21v-28h-277v28q63 4 78 21t15 80v761q0 20 -6 25t-27 5h-358q-23 0 -28.5 -5.5t-5.5 -26.5v-759q0 -63 14 -80t72 -21v-28h-268v28q58 4 72 21t14 80v709q0 63 -14.5 79.5t-75.5 21.5v28h798v-28z
 /></g> <g transform=matrix(.012,-0,0,-.012,41.525,11.963)><use xlink:href=#x1D45A/></g><g transform=matrix(.012,-0,0,-.012,41.525,24.712)><path id=x1D457 d=M394 607q0 -25 -15.5 -43t-39.5 -18q-19 0 -32 13t-13 35t16 41.5t40 19.5q20 0 32 -13.5t12 -34.5zM360 349l-57 -283q-38 -188 -122 -263q-72 -64 -114 -64q-16 0 -30 10.5t-14 22.5q0 24 21 45q15 13 26 6q30 -20 54 -20q41 0 65 95q37 147 86 415q6 34 6.5 42.5
t-5.5 11.5q-18 0 -79 -54l-14 25q48 48 96.5 79t71.5 31q29 0 9 -99z /></g><g transform=matrix(.012,-0,0,-.012,46.558,24.712)><path id=x3D d=M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z /></g><g transform=matrix(.012,-0,0,-.012,53.543,24.712)><use xlink:href=#x31/></g> <g transform=matrix(.017,-0,0,-.017,59.875,20.237)><use xlink:href=#x1D714/></g> <g transform=matrix(.012,-0,0,-.012,70.725,8.425)><use xlink:href=#x1D45D/></g><g transform=matrix(.012,-0,0,-.012,77.781,8.425)><path id=x2F d=M368 703l-264 -866h-60l265 866h59z /></g><g transform=matrix(.012,-0,0,-.012,82.683,8.425)><use xlink:href=#x1D45D/></g> <g transform=matrix(.008,-0,0,-.008,89.738,11.913)><use xlink:href=#x1D457/></g> <g transform=matrix(.012,-0,0,-.012,70.725,25.325)><use xlink:href=#x1D457/></g> <g transform=matrix(.017,-0,0,-.017,94.588,20.237)><use xlink:href=#x29/></g> </svg>, we obtained that they are also bounded from <svg style=vertical-align:-3.39066pt;width:180.6375px; id=M4 height=20.975 version=1.1 viewBox=0 0 180.6375 20.975 width=180.6375 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,16.7)><use xlink:href=#x1D43F/></g> <g transform=matrix(.012,-0,0,-.012,9.763,8.537)><use xlink:href=#x1D45D/></g> <g transform=matrix(.008,-0,0,-.008,16.825,12.012)><use xlink:href=#x31/></g> <g transform=matrix(.012,-0,0,-.012,21.525,8.537)><use xlink:href=#x2C/></g><g transform=matrix(.012,-0,0,-.012,24.238,8.537)><use xlink:href=#x1D458/></g> <g transform=matrix(.017,-0,0,-.017,30.85,16.7)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,36.732,16.7)><use xlink:href=#x1D714/></g> <g transform=matrix(.012,-0,0,-.012,47.575,20.775)><use xlink:href=#x1D456/></g> <g transform=matrix(.017,-0,0,-.017,51.45,16.7)><use xlink:href=#x29/></g><g transform=matrix(.017,-0,0,-.017,61.088,16.7)><use xlink:href=#xD7/></g><g transform=matrix(.017,-0,0,-.017,77.679,16.7)><use xlink:href=#x22EF/></g><g transform=matrix(.017,-0,0,-.017,100.78,16.7)><use xlink:href=#xD7/></g><g transform=matrix(.017,-0,0,-.017,114.532,16.7)><use xlink:href=#x1D43F/></g> <g transform=matrix(.012,-0,0,-.012,124.25,8.537)><use xlink:href=#x1D45D/></g> <g transform=matrix(.008,-0,0,-.008,131.312,12.012)><use xlink:href=#x1D45A/></g> <g transform=matrix(.012,-0,0,-.012,138.638,8.537)><use xlink:href=#x2C/></g><g transform=matrix(.012,-0,0,-.012,141.35,8.537)><use xlink:href=#x1D458/></g> <g transform=matrix(.017,-0,0,-.017,147.962,16.7)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,153.844,16.7)><use xlink:href=#x1D714/></g> <g transform=matrix(.012,-0,0,-.012,164.688,20.775)><use xlink:href=#x1D45A/></g> <g transform=matrix(.017,-0,0,-.017,174.7,16.7)><use xlink:href=#x29/></g> </svg> to <svg style=vertical-align:-6.57922pt;width:109.225px; id=M5 height=28.487499 version=1.1 viewBox=0 0 109.225 28.487499 width=109.225 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,20.237)><use xlink:href=#x1D43F/></g> <g transform=matrix(.012,-0,0,-.012,9.763,12.075)><use xlink:href=#x1D45D/></g><g transform=matrix(.012,-0,0,-.012,16.818,12.075)><use xlink:href=#x2C/></g><g transform=matrix(.012,-0,0,-.012,19.531,12.075)><use xlink:href=#x1D458/></g> <g transform=matrix(.017,-0,0,-.017,26.15,20.237)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,32.032,20.237)><use xlink:href=#x220F/></g> <g transform=matrix(.012,-0,0,-.012,50.225,11.963)><use xlink:href=#x1D45A/></g><g transform=matrix(.012,-0,0,-.012,50.225,24.712)><use xlink:href=#x1D457/></g><g transform=matrix(.012,-0,0,-.012,55.258,24.712)><use xlink:href=#x3D/></g><g transform=matrix(.012,-0,0,-.012,62.243,24.712)><use xlink:href=#x31/></g> <g transform=matrix(.017,-0,0,-.017,68.575,20.237)><use xlink:href=#x1D714/></g> <g transform=matrix(.012,-0,0,-.012,79.425,8.425)><use xlink:href=#x1D45D/></g><g transform=matrix(.012,-0,0,-.012,86.481,8.425)><use xlink:href=#x2F/></g><g transform=matrix(.012,-0,0,-.012,91.383,8.425)><use xlink:href=#x1D45D/></g> <g transform=matrix(.008,-0,0,-.008,98.438,11.913)><use xlink:href=#x1D457/></g> <g transform=matrix(.012,-0,0,-.012,79.425,25.325)><use xlink:href=#x1D457/></g> <g transform=matrix(.017,-0,0,-.017,103.275,20.237)><use xlink:href=#x29/></g> </svg> with <svg style=vertical-align:-0.1638pt;width:63.837502px; id=M6 height=12.4375 version=1.1 viewBox=0 0 63.837502 12.4375 width=63.837502 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,12.162)><path id=x30 d=M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
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