Sobolev–Lorentz spaces in the Euclidean setting and counterexamples

Costea, Şerban

doi:10.1016/j.na.2017.01.001

Cited by 9 publications

(39 citation statements)

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“…We studied the Sobolev-Lorentz spaces and their associated capacities extensively in our previous work. In this paper we extend some of the previous results obtained in our book [6] and in our papers [5] and [7]. In [6] we studied the Sobolev-Lorentz spaces and the associated Sobolev-Lorentz capacities in the Euclidean setting for n ≥ 2.…”

Section: Introductionsupporting

confidence: 61%

“…See (3). Thus, Section 6 together with our paper [7] (see [7,Theorems 3.5,4.3,4.13 and 5.6]) reinforce the fact that for every n ≥ 2 integer and for every Ω ⊂ R n open, the space H 1,(n,1) loc (Ω) is the largest Sobolev-Lorentz space defined on Ω for which each function has a version in C(Ω).…”

Section: Introductionmentioning

confidence: 58%

“…The restriction on n there as well as in [5] was due to the fact that we studied the n, q capacity for n > 1. In our recent paper [7] we studied the Sobolev-Lorentz spaces in the Euclidean setting for n ≥ 1. There we extended to the case n = 1 many of the results on Sobolev-Lorentz spaces obtained in [4] and [6] for n ≥ 2.…”

Section: Introductionmentioning

confidence: 99%

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Sobolev–Lorentz capacity and its regularity in the Euclidean setting

Costea¹

2019

Ann. Acad. Sci. Fenn. Math.

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This paper studies the Sobolev-Lorentz capacity and its regularity in the Euclidean setting for n ≥ 1 integer. We extend here our previous results on the Sobolev-Lorentz capacity obtained for n ≥ 2.Moreover, for n ≥ 2 integer we obtain a few new results concerning the n, 1 relative and global capacities. Specifically, we obtain sharp estimates for the n, 1 relative capacity of the concentric condensers (B(0, r), B(0, 1)) for all r in [0, 1). As a consequence we obtain the exact value of the n, 1 capacity of a point relative to all its bounded open neighborhoods from R n when n ≥ 2. These new sharp estimates concerning the n, 1 relative capacity improve some of our previous results. We also obtain a new result concerning the n, 1 global capacity. Namely, we show that this aforementioned constant is also the value of the n, 1 global capacity of any point from R n , where n ≥ 2 is integer.Computing the aforementioned exact value of the n, 1 relative capacity of a point with respect to all its bounded open neighborhoods from R n allows us to give a new prove of the embedding H

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Section: Introductionsupporting

confidence: 61%

Section: Introductionmentioning

confidence: 58%

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Sobolev–Lorentz capacity and its regularity in the Euclidean setting

Costea¹

2019

Ann. Acad. Sci. Fenn. Math.

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“…In the case q = ∞ the functional setting is somewhat delicate, as no Meyer-Serrin type result holds for the corresponding homogeneous spaces. Thus, let us define for 1 ≤ p < ∞ and 1 ≤ q ≤ ∞ the space [10] and Section 2 for more details. Then, surprisingly, we establish the equivalence between (A p,∞ ) and (H p ).…”

Section: Resultsmentioning

confidence: 99%

“…Let us suppose the inequality (A p,q ) is attained at some q < ∞. Following the lines of Section 3, we have at least one radially decreasing maximizer u ∈ D 1 L p,q (R n ), namely a function u such that (10) u p * ,q = p n − p ω (the last inequality is trivial: set v ≡ u, f ≡ |∇u|). It is known that for any f ≥ 0, f ∈ L p,q (Ω) there exists a maximal nonnegative sub-solution v ∈ W 1 L p,q 0 (Ω) of the problem (11) |∇v| ≤ f (see [21] Prop.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Equivalent and attained version of Hardy's inequality in Rn

Cassani

Ruf

Tarsi

2018

Journal of Functional Analysis

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Regularity theory and Green’s function for elliptic equations with lower order terms in unbounded domains

Mourgoglou

2023

Calc. Var.

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We consider elliptic operators in divergence form with lower order terms of the form $$Lu = -{{\textrm{div}}}(A \cdot \nabla u + b u ) - c \cdot \nabla u - du$$ L u = - div ( A · ∇ u + b u ) - c · ∇ u - d u , in an open set $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n , $$n \ge 3$$ n ≥ 3 , with possibly infinite Lebesgue measure. We assume that the $$n \times n$$ n × n matrix A is uniformly elliptic with real, merely bounded and possibly non-symmetric coefficients, and either $$b, c \in L^{n,\infty }_{\text {loc}}({\Omega })$$ b , c ∈ L loc n , ∞ ( Ω ) and $$d \in L_{\text {loc}}^{\frac{n}{2}, \infty }(\Omega )$$ d ∈ L loc n 2 , ∞ ( Ω ) , or $$|b|^2, |c|^2, |d| \in \mathcal {K}_{\text {loc}}(\Omega )$$ | b | 2 , | c | 2 , | d | ∈ K loc ( Ω ) , where $$\mathcal {K}_{\text {loc}}(\Omega )$$ K loc ( Ω ) stands for the local Stummel–Kato class. Let $${\mathcal {K}_{\text {Dini}}}(\Omega )$$ K Dini ( Ω ) be a variant of $$\mathcal {K}(\Omega )$$ K ( Ω ) satisfying a Carleson-Dini-type condition. We develop a De Giorgi/Nash/Moser theory for solutions of $$Lu = f - {{\textrm{div}}}g$$ L u = f - div g , where f and $$|g|^2 \in {\mathcal {K}_{\text {Dini}}}(\Omega )$$ | g | 2 ∈ K Dini ( Ω ) if, for $$q \in [n, \infty )$$ q ∈ [ n , ∞ ) , any of the following assumptions holds: (i) $$|b|^2, |d| \in {\mathcal {K}_{\text {Dini}}}(\Omega )$$ | b | 2 , | d | ∈ K Dini ( Ω ) and either $$c \in L^{n,q}_{\text {loc}}(\Omega )$$ c ∈ L loc n , q ( Ω ) or $$|c|^2 \in \mathcal {K}_{\text {loc}}(\Omega )$$ | c | 2 ∈ K loc ( Ω ) ; (ii) $${{\textrm{div}}}b +d \le 0$$ div b + d ≤ 0 and either $$b+c \in L^{n,q}_{\text {loc}}(\Omega )$$ b + c ∈ L loc n , q ( Ω ) or $$|b+c|^2 \in \mathcal {K}_{\text {loc}}(\Omega )$$ | b + c | 2 ∈ K loc ( Ω ) ; (iii) $$-{{\textrm{div}}}c + d \le 0$$ - div c + d ≤ 0 and $$|b+c|^2 \in {\mathcal {K}_{\text {Dini}}}(\Omega )$$ | b + c | 2 ∈ K Dini ( Ω ) . We also prove a Wiener-type criterion for boundary regularity. Assuming global conditions on the coefficients, we show that the variational Dirichlet problem is well-posed and, assuming $$-{{\textrm{div}}}c +d \le 0$$ - div c + d ≤ 0 , we construct the Green’s function associated with L satisfying quantitative estimates. Under the additional hypothesis $$|b+c|^2 \in \mathcal {K}'(\Omega )$$ | b + c | 2 ∈ K ′ ( Ω ) , we show that it satisfies global pointwise bounds and also construct the Green’s function associated with the formal adjoint operator of L. An important feature of our results is that all the estimates are scale invariant and independent of $$\Omega $$ Ω , while we do not assume smallness of the norms of the coefficients or coercivity of the associated bilinear form.

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Sobolev–Lorentz spaces in the Euclidean setting and counterexamples

Abstract: Abstract. This paper studies the inclusions between different Sobolev-Lorentz spaces W

Cited by 9 publications

References 25 publications

Sobolev–Lorentz capacity and its regularity in the Euclidean setting

Sobolev–Lorentz capacity and its regularity in the Euclidean setting

Equivalent and attained version of Hardy's inequality in Rn

Regularity theory and Green’s function for elliptic equations with lower order terms in unbounded domains

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