2012
DOI: 10.1016/j.jmaa.2012.06.054
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Fractional Gagliardo–Nirenberg and Hardy inequalities under Lorentz norms

Abstract: Keywords:Fractional Laplacian Gagliardo-Nirenberg inequality Sobolev inequality Logarithmic Sobolev inequality Hardy inequality a b s t r a c tIn this paper, we establish the Gagliardo-Nirenberg inequality under Lorentz norms for fractional Laplacian. Based on special cases of this inequality under Lebesgue norms, we prove the L p -logarithmic Gagliardo-Nirenberg and Sobolev inequalities. Motivated by the L 2 -logarithmic Sobolev inequality, we obtain a fractional logarithmic Sobolev trace inequality in terms … Show more

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Cited by 80 publications
(44 citation statements)
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“…By the Fractional Gagliardo-Nirenberg inequality ( [29],corollary 2.3]) and the definition of the norm in X , we know that…”
Section: Resultsmentioning
confidence: 99%
“…By the Fractional Gagliardo-Nirenberg inequality ( [29],corollary 2.3]) and the definition of the norm in X , we know that…”
Section: Resultsmentioning
confidence: 99%
“…In the above statement D(R d ) denotes the space of smooth functions with compact support, but it is simple to extend it by density to the space w ∈H α 2 (R d ) : [26,Theorem 2.1] for details. When p = q 2 = d d−α , then ϑ = 1 and C GNS = S d,α .…”
Section: A Fractional Gagliardo-nirenberg-sobolev Inequalitymentioning
confidence: 99%
“…It should be mentioned that there is increasing literature devoted to the study of improved versions of the Sobolev-Gagliardo-Nirenberg and related inequalities for their own sake. Besides the above mentioned articles [9] and [16], we can mention [1,4,11,12,13,19], among others.…”
Section: Introductionmentioning
confidence: 99%