{"id":18566,"date":"2023-11-22T10:47:06","date_gmt":"2023-11-22T02:47:06","guid":{"rendered":"https:\/\/virtual.math.ntnu.edu.tw\/?p=18566"},"modified":"2023-11-27T17:59:17","modified_gmt":"2023-11-27T09:59:17","slug":"lecture20231213","status":"publish","type":"post","link":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/2023\/11\/22\/lecture20231213\/","title":{"rendered":"[\u5c08\u984c\u6f14\u8b1b] <\/span>\u301012\u670813\u65e5\u3011David Cruz-Uribe \/ Norm inequalities for linear and multilinear singular integrals on weighted and variable exponent Hardy spaces"},"content":{"rendered":"\t\t
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Norm inequalities for linear and multilinear singular integrals on weighted and variable exponent Hardy spaces<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\u6642\u3000\u9593\uff1a2023-12-13 14:00 (\u661f\u671f\u4e09) \/ \u5730\u3000\u9ede\uff1aS101 \/ \u8336\u3000\u6703\uff1aS205 (13:30)<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t

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David Cruz-Uribe<\/div>
University of Alabama<\/div>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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I will discuss a (relatively) new approach to norm inequalities in the weighted and variable exponent Hardy spaces. The weighted Hardy spaces H<\/mi>p<\/mi><\/msup>(<\/mo>w<\/mi>)<\/mo><\/mrow>H^p(w)<\/annotation><\/semantics><\/math><\/span><\/span>, p<\/mi>><\/mo>0<\/mn><\/mrow>p>0<\/annotation><\/semantics><\/math><\/span><\/span>, where w<\/mi><\/mrow>w<\/annotation><\/semantics><\/math><\/span><\/span> is a Muckenhoupt weight, were first considered by Stromberg and Torchinsky in the 1980s. The variable Lebesgue space L<\/mi>p<\/mi>(<\/mo>\u22c5<\/mo>)<\/mo><\/mrow><\/msup><\/mrow>L^{p(\\cdot)}<\/annotation><\/semantics><\/math><\/span><\/span> is, intuitively, a classical Lebesgue space with the constant exponent p<\/mi><\/mrow>p<\/annotation><\/semantics><\/math><\/span><\/span> replaced by an exponent function p<\/mi>(<\/mo>\u22c5<\/mo>)<\/mo><\/mrow>p(\\cdot)<\/annotation><\/semantics><\/math><\/span><\/span>. They have been studied extensively for the last 30 years. The corresponding variable Hardy spaces H<\/mi>p<\/mi>(<\/mo>\u22c5<\/mo>)<\/mo><\/mrow><\/msup><\/mrow>H^{p(\\cdot)}<\/annotation><\/semantics><\/math><\/span><\/span> were introduced by me and Li-An Wang and independently by Nakai and Sawano.<\/span><\/p>

We give inter-related conditions on a Calderon-Zygmund singular integral operator T<\/mi><\/mrow>T<\/annotation><\/semantics><\/math><\/span><\/span>, a weight w<\/mi><\/mrow>w<\/annotation><\/semantics><\/math><\/span><\/span>, and an exponent p<\/mi>(<\/mo>\u22c5<\/mo>)<\/mo><\/mrow>p(\\cdot)<\/annotation><\/semantics><\/math><\/span><\/span> for T<\/mi><\/mrow>T<\/annotation><\/semantics><\/math><\/span><\/span> to satisfy estimates of the formT<\/mi>:<\/mo>H<\/mi>p<\/mi><\/msup>(<\/mo>w<\/mi>)<\/mo>\u2192<\/mo>L<\/mi>p<\/mi><\/msup>(<\/mo>w<\/mi>)<\/mo>,<\/mo><\/mspace>T<\/mi>:<\/mo>H<\/mi>p<\/mi>(<\/mo>\u22c5<\/mo>)<\/mo><\/mrow><\/msup>\u2192<\/mo>L<\/mi>p<\/mi>(<\/mo>\u22c5<\/mo>)<\/mo><\/mrow><\/msup>.<\/mi><\/mrow> T : H^p(w) \\rightarrow L^p(w), \\qquad T: H^{p(\\cdot)} \\rightarrow L^{p(\\cdot)}.<\/annotation><\/semantics><\/math><\/span><\/span><\/span> <\/span><\/p>

Some of our results were known for convolution type singular integrals, but we give new and simpler proofs and give extensions to non-convolution type operators. Our proofs depend very heavily on three tools: atomic decompositions of the Hardy spaces, vector-valued inequalities, and the Rubio de Francia theory of extrapolation.<\/span><\/p>

We will also discuss generalizations of these results to the bilinear setting, where we prove norm inequalities of the form
T<\/mi>:<\/mo>H<\/mi>p<\/mi>1<\/mn><\/msub><\/msup>(<\/mo>w<\/mi>1<\/mn><\/msub>)<\/mo>\u00d7<\/mo>H<\/mi>p<\/mi>2<\/mn><\/msub><\/msup>(<\/mo>w<\/mi>2<\/mn><\/msub>)<\/mo>\u2192<\/mo>L<\/mi>p<\/mi><\/msup>(<\/mo>w<\/mi>)<\/mo>,<\/mo><\/mrow> T: H^{p_1}(w_1) \\times H^{p_2}(w_2) \\rightarrow L^p(w), <\/annotation><\/semantics><\/math><\/span><\/span><\/span>\u00a0
where T<\/mi><\/mrow>T<\/annotation><\/semantics><\/math><\/span><\/span> is a bilinear Calder\\’on-Zygmund singular integral operator, p<\/mi>,<\/mo>\u2009p<\/mi>1<\/mn><\/msub>,<\/mo>\u2009p<\/mi>2<\/mn><\/msub>><\/mo>0<\/mn><\/mrow>p,\\,p_1,\\,p_2>0<\/annotation><\/semantics><\/math><\/span><\/span> and
1<\/mn>p<\/mi>1<\/mn><\/msub><\/mfrac>+<\/mo>1<\/mn>p<\/mi>2<\/mn><\/msub><\/mfrac>=<\/mo>1<\/mn>p<\/mi><\/mfrac>,<\/mo><\/mrow> \\frac{1}{p_1}+\\frac{1}{p_2} = \\frac{1}{p}, <\/annotation><\/semantics><\/math><\/span><\/span><\/span>\u00a0
and the weights w<\/mi>,<\/mo>\u2009w<\/mi>1<\/mn><\/msub>,<\/mo>\u2009w<\/mi>2<\/mn><\/msub><\/mrow>w,\\,w_1,\\,w_2<\/annotation><\/semantics><\/math><\/span><\/span> are Muckenhoupt weights. We also consider norm inequalities of the form
T<\/mi>:<\/mo>H<\/mi>p<\/mi>1<\/mn><\/msub>(<\/mo>\u22c5<\/mo>)<\/mo><\/mrow><\/msup>\u00d7<\/mo>H<\/mi>p<\/mi>2<\/mn><\/msub>(<\/mo>\u22c5<\/mo>)<\/mo><\/mrow><\/msup>\u2192<\/mo>L<\/mi>p<\/mi>(<\/mo>\u22c5<\/mo>)<\/mo><\/mrow><\/msup>.<\/mi><\/mrow> T: H^{p_1(\\cdot)}\\times H^{p_2(\\cdot)}\\rightarrow L^{p(\\cdot)}. <\/annotation><\/semantics><\/math><\/span>
<\/span><\/span><\/span><\/p>

This is joint work with Kabe Moen and Hanh Nguyen of the University of Alabama.<\/span><\/p>

\u66f4\u591a\u8cc7\u8a0a
https:\/\/arxiv.org\/abs\/1902.01519<\/a>
<\/strong><\/span><\/p>

\u500b\u4eba\u7db2\u9801\u00a0<\/span>
https:\/\/math.ua.edu\/people\/david-cruz-uribe\/<\/a><\/span>
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\u53cb\u5584\u5217\u5370<\/a><\/p>","protected":false},"excerpt":{"rendered":"

Norm inequalities for linear and multilinear singular i […]<\/p>\n","protected":false},"author":23,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[1,124],"tags":[],"_links":{"self":[{"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts\/18566"}],"collection":[{"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/users\/23"}],"replies":[{"embeddable":true,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/comments?post=18566"}],"version-history":[{"count":8,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts\/18566\/revisions"}],"predecessor-version":[{"id":18611,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts\/18566\/revisions\/18611"}],"wp:attachment":[{"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/media?parent=18566"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/categories?post=18566"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/tags?post=18566"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}