Almost everywhere convergence of the convolution type Laguerre expansions

2025/04/09 by Wei, Longben
#35P10 #42B15 #42B25 #42C10 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2504.06713

Abstract

For a fixed d-tuple α=(α₁,...,α_d)∈(-1,∞)^d, consider the product space ℝ₊^d:=(0,∞)^d equipped with Euclidean distance \arrowvert ⋅ \arrowvert and the measure dμ_α(x)=x₁^2α₁+1⋅⋅⋅ x_d^α_ddx₁⋅⋅⋅ dx_d. We consider the Laguerre operator L_α=-Δ+∑_i=1^d(2α_j+1)/(x_j)(d)/(dx_j)+\arrowvert x\arrowvert² which is a compact, positive, self-adjoint operator on L²(ℝ₊^d,dμ_α(x)). In this paper, we study almost everywhere convergence of the Bochner-Riesz means associated with L_α which is defined by S_R^λ(L_α)f(x)=∑_n=0^∞(1-(e_n)/(R²))₊^λP_nf(x). Here e_n is n-th eigenvalue of L_α, and P_nf(x) is the n-th Laguerre spectral projection operator. This corresponds to the convolution-type Laguerre expansions introduced in Thangavelu's lecture \citeTS3. For 2≤ p<∞, we prove that lim_R→∞ S_R^λ(L_α)f=f -a.e. for all f∈ L^p(ℝ₊^d,dμ_α(x)), provided that λ>λ(α,p)/2, where λ(α,p)=max\2(\arrowvertα\arrowvert₁+d)(1/2-1/p)-1/2,0\, and \arrowvertα\arrowvert₁:=∑_j=1^dα_j. Conversely, if 2\arrowvertα\arrowvert₁+2d>1, we will show the convergence generally fails if λ<λ(α,p)/2 in the sense that there is an f∈ L^p(ℝ₊^d,dμ_α(x)) for (4\arrowvertα\arrowvert₁+4d)/(2\arrowvertα\arrowvert₁+2d-1)< p such that the convergence fails. When 2\arrowvertα\arrowvert₁+2d≤1, our results show that a.e. convergence holds for f∈ L^p(ℝ₊^d,dμ_α(x)) with p≥ 2 whenever λ>0.

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