Best Constant Inequalities Involving the Analytic and Co-Analytic Projection
Best Constant Inequalities Involving the Analytic and Co-Analytic Projection
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DOI：
10.1007/978-3-0346-0158-0_15
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
B. Hollenbeck;I. Verbitsky
中科院分区：
文献类型：
--
作者：
B. Hollenbeck;I. Verbitsky
LetP+denote the Riesz projection andP−=I−P+ denote the co-analytic projection where I is the identity operator. We prove $$ \left\| {\max (\left| {P_ + f} \right|,\left| {P_ - f} \right|)} \right\|_{L^p (T)} \leqslant \csc \frac{\pi } {p}\left\| f \right\|_{L^p (T)} , 1 < p < \infty , $$ wheref∈Lp(T) is a complex-valued function, and the constantp p is sharp. Our proof is based on an explicit construction of a plurisubharmonic minorant for the functiononC2. More generally, we discuss the best constant problem for the inequality $$ \left\| {(\left| {P_ + f} \right|^s ,\left| {P_ - f} \right|^s )^{\tfrac{1} {s}} } \right\|_{L^p (T)} \leqslant C(p,s)\left\| f \right\|_{L^p (T)} , 1 < p < \infty , $$ where 0<s<∞, which may serve as a model problem for some vectorvalued inequalities, where the method of plurisubharmonic minorants seems to be promising.
LetP+denote the Riesz projection andP−=I−P+ denote the co-analytic projection where I is the identity operator. We prove $$ \left\| {\max (\left| {P_ + f} \right|,\left| {P_ - f} \right|)} \right\|_{L^p (T)} \leqslant \csc \frac{\pi } {p}\left\| f \right\|_{L^p (T)} , 1 < p < \infty , $$ wheref∈Lp(T) is a complex-valued function, and the constantp p is sharp. Our proof is based on an explicit construction of a plurisubharmonic minorant for the functiononC2. More generally, we discuss the best constant problem for the inequality $$ \left\| {(\left| {P_ + f} \right|^s ,\left| {P_ - f} \right|^s )^{\tfrac{1} {s}} } \right\|_{L^p (T)} \leqslant C(p,s)\left\| f \right\|_{L^p (T)} , 1 < p < \infty , $$ where 0<s<∞, which may serve as a model problem for some vectorvalued inequalities, where the method of plurisubharmonic minorants seems to be promising.