[Paper Review] Product operators on mixed norm spaces

Wayne Grey, Gord Sinnamon|arXiv (Cornell University)|Feb 29, 2016

Advanced Harmonic Analysis Research11 references3 citations

TL;DR

This paper establishes sharp inequalities for product operators on mixed norm Lebesgue spaces using only the Lebesgue indices and factor operator bounds, providing a unified framework for analyzing multilinear operators. It offers an elementary proof of the multivariate Young’s inequality in mixed norm spaces via recursive application of Minkowski’s integral inequality and convolution estimates.

ABSTRACT

Inequalities for product operators on mixed norm Lebesgue spaces and permuted mixed norm Lebesgue spaces are established. They depend only on inequalities for the factors and on the Lebesgue indices involved. Inequalities for the bivariate Laplace transform are given to illustrate the method. Also, an elementary proof is presented for an $n$-variable Young's inequality in mixed norm spaces.

Motivation & Objective

• To extend techniques from embedding theory in mixed norm spaces to general product operators beyond the identity operator.
• To derive norm inequalities for product operators based solely on the Lebesgue indices and individual factor operator bounds.
• To provide a systematic treatment of permuted mixed norms, which arise when the order of integration differs across spaces.
• To offer a new, elementary proof of the multivariate Young’s inequality in mixed norm spaces using recursive Minkowski and convolution estimates.
• To demonstrate the method via inequalities for the bivariate Laplace transform and to generalize results to locally compact unimodular groups.

Proposed method

• Define product operators as integral operators with separable kernels $ Kf(x_1,x_2) = \iint k_1(x_1,t_1)k_2(x_2,t_2)f(t_1,t_2) \, d(\lambda_1 \times \lambda_2)(t_1,t_2) $.
• Express the operator norm of $ K $ in terms of the individual operator norms $ \|K_j\|_{L^{p_j}_{\lambda_j} \to L^{r_j}_{\mu_j}} $, using the factorization $ K = K_1 \otimes K_2 $.
• Apply Minkowski’s integral inequality in mixed norm form: $ \|f\|_{L^{(p_1,p_2)}_{\lambda_1 \times \lambda_2}} \leq \|f\|_{L^{(p_2,p_1)}_{\lambda_2 \times \lambda_1}} $ for non-negative functions.
• Use recursive application of Minkowski’s inequality and Young’s inequality to bound the mixed norm of the convolution $ f*g $ in terms of $ \|f\|_{L^P} \|g\|_{L^Q} $.
• Establish the key inequality $ \|f*g\|_{L^R} \leq \|f\|_{L^P} \|g\|_{L^Q} $ under the condition $ \frac{1}{p_j} + \frac{1}{q_j} = \frac{1}{r_j} + 1 $ for each $ j $.
• Handle general $ L^p $ functions by approximating them with bounded, integrable functions and applying the dominated convergence theorem to extend the inequality to non-negative and signed functions.

Experimental results

Research questions

• RQ1 How can inequalities for product operators on mixed norm spaces be derived from the norms of their individual factors?
• RQ2 What conditions on Lebesgue indices ensure boundedness of product operators between mixed norm Lebesgue spaces?
• RQ3 Can the classical multivariate Young’s inequality in mixed norm spaces be proven using only Minkowski’s integral inequality and recursive estimation?
• RQ4 How do permuted mixed norms affect the boundedness and norm estimates of product operators?
• RQ5 To what extent can the method be generalized to convolution operators on locally compact unimodular groups?

Key findings

• The paper establishes that the operator norm of a product operator $ K $ on mixed norm spaces depends only on the Lebesgue indices $ p_1, p_2, r_1, r_2 $ and the individual operator norms $ C_1, C_2 $ of the factors.
• An elementary proof of the multivariate Young’s inequality in mixed norm spaces is provided, showing $ \|f*g\|_{L^R} \leq \|f\|_{L^P} \|g\|_{L^Q} $ under the condition $ \frac{1}{p_j} + \frac{1}{q_j} = \frac{1}{r_j} + 1 $ for each $ j $.
• The proof relies on recursive application of Minkowski’s integral inequality and convolution estimates, with intermediate functions $ f_j, g_j $ defined via $ L^{p_j} $-norms over successive variables.
• The method extends to cases where some indices are infinite, with the supremum norm used in place of $ L^\infty $-norms, though the proof requires modification.
• The framework applies to the bivariate Laplace transform, yielding explicit inequalities based on the same principles.
• The results generalize to convolution on finite products of locally compact unimodular groups, maintaining the same norm inequality structure.