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Best constants in inequalities involving analytic and co-analytic projections and Riesz theorem for various function spaces
Functional Analysis · Submitted 18 Dec 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We find the accurate estimates for $p\geq 2$ and $0<s\leq p$, thus significantly improving results from \cite{KALAJ.TAMS} where it is considered for $s=2$ and $1<p<\infty$.
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\begin{abstract} Let $P\pm$ be the Riesz's projection operator and let $P_-= I - P_+$. We consider estimates of the expression $\|( |P_ + f | ^s + |P_- f |^s) ^{\frac{1}{s}}\|_{L^p (\mathbf{T})}$ in terms of Lebesgue $p$-norm of the function $f \in L^p(\mathbf{T})$. We find the accurate estimates for $p\geq 2$ and $0<s\leq p$, thus significantly improving results from \cite{KALAJ.TAMS} where it is considered for $s=2$ and $1<p<\infty$. Interestingly, for this range of $s$ there holds the appropriate vector-valued inequality with the same constant. Also, we obtain the right asymptotic of the constants for large $s$. This proves the conjecture of Hollenbeck and Verbitsky on the Riesz projection operator in some cases. As a consequence of inequalities we have proved in the paper we get Riesz-type theorems on conjugate harmonic functions for various function spaces. In particular, slightly general version of Stout's theorem for Lumer Hardy spaces is obtained by a new approach. \end{abstract}
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | \begin{abstract} Let $P\pm$ be the Riesz's projection operator and let $P_-= I - P_+$. | background 1.00 | 0.11 | 0.02 | asserted 0.77 |
| s1 | We consider estimates of the expression $\|( |P_ + f | ^s + |P_- f |^s) ^{\frac{1}{s}}\|_{L^p (\mathbf{T})}$ in terms of Lebesgue $p$-norm of the function $f \in L^p(\mathbf{T})$. | contribution 0.81 | 0.69 | 0.06 | asserted 0.77 |
| s2 | We find the accurate estimates for $p\geq 2$ and $0<s\leq p$, thus significantly improving results from \cite{KALAJ.TAMS} where it is considered for $s=2$ and $1<p<\infty$. | result 0.80 | 0.95 | 0.53 | asserted 1.00 |
| s3 | Interestingly, for this range of $s$ there holds the appropriate vector-valued inequality with the same constant. | result 0.90 | 0.79 | 0.08 | asserted 1.00 |
| s4 | Also, we obtain the right asymptotic of the constants for large $s$. | result 0.97 | 0.96 | 0.08 | asserted 1.00 |
| s5 | This proves the conjecture of Hollenbeck and Verbitsky on the Riesz projection operator in some cases. | interpretation 0.80 | 0.85 | 0.23 | asserted 1.00 |
| s6 | As a consequence of inequalities we have proved in the paper we get Riesz-type theorems on conjugate harmonic functions for various function spaces. | result 0.53 | 0.94 | 0.22 | asserted 1.00 |
| s7 | In particular, slightly general version of Stout's theorem for Lumer Hardy spaces is obtained by a new approach. \end{abstract} | result 0.89 | 0.93 | 0.14 | asserted 1.00 |
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Original abstract
\begin{abstract} Let $P\pm$ be the Riesz's projection operator and let $P_-= I - P_+$. We consider estimates of the expression $\|( |P_ + f | ^s + |P_- f |^s) ^{\frac{1}{s}}\|_{L^p (\mathbf{T})}$ in terms of Lebesgue $p$-norm of the function $f \in L^p(\mathbf{T})$. We find the accurate estimates for $p\geq 2$ and $0<s\leq p$, thus significantly improving results from \cite{KALAJ.TAMS} where it is considered for $s=2$ and $1<p<\infty$. Interestingly, for this range of $s$ there holds the appropriate vector-valued inequality with the same constant. Also, we obtain the right asymptotic of the constants for large $s$. This proves the conjecture of Hollenbeck and Verbitsky on the Riesz projection operator in some cases. As a consequence of inequalities we have proved in the paper we get Riesz-type theorems on conjugate harmonic functions for various function spaces. In particular, slightly general version of Stout's theorem for Lumer Hardy spaces is obtained by a new approach. \end{abstract}
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