Lower estimates for the norm and the Kuratowski measure of moncompactness of Wiener-Hopf type operators

2025/09/24 by Karlovych, Oleksiy, Shargorodsky, Eugene
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2509.20296

Abstract

Let X(ℝⁿ) be a Banach function space and Ω⊆ℝⁿ be a measurable set of positive measure. For a Fourier mutliplier a on X(ℝⁿ), consider the Wiener-Hopf type operator W_Ω(a):=r_ΩF^-1aF e_Ω, where F^{± 1} are the Fourier transforms, r_Ω is the operator of restriction from ℝⁿ to Ω and e_Ω is the operator of extension by zero from Ω to ℝⁿ. Let X₂(Ω) be the closure of L²(Ω)∩ X(Ω) in X(Ω). We show that if X(Ω) satisfies the so-called weak doubling property, then ‖a‖_{L^∞(ℝⁿ)} ≤ ‖W_Ω(a)‖_{B(X2(Ω),X(Ω))}. Further, we prove that if X(Ω) satisfies the so-called separated doubling property, then the Kuratowski measure of noncompactness of W_Ω(a) admits the following lower estimate: (1)/(2)‖a‖_{L^∞(ℝⁿ)} ≤ ‖W_Ω(a)‖_{B(X2(Ω),X(Ω)),κ}. These results are specified to the case of variable Lebesgue spaces L^p(⋅)(C,w) with Muckenhoupt type weights w over open cones C⊆ℝⁿ with the vertex at the origin.

Citations

• Maximal Noncompactness of Wiener-Hopf Operators
• Fredholm criteria for Wiener-Hopf operators with continuous symbols acting on some Banach function spaces
• When does the norm of a Fourier multiplier dominate its L^∞ norm?
• The Cauchy Singular Integral Operator on Weighted Variable Lebesgue\n Spaces

Related

• Maximal Noncompactness of Wiener-Hopf Operators
• When does the norm of a Fourier multiplier dominate its L^∞ norm?
• Fredholm criteria for Wiener-Hopf operators with continuous symbols acting on some Banach function spaces
• Norm estimates for a broad class of modulation spaces, and continuity of Fourier type operators
• Equivalence of Sobolev norms with respect to weighted Gaussian measures