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MINIMIZING PROBLEMS FOR THE HARDY-SOBOLEV TYPE INEQUALITY WITH THE SINGULARITY ON THE BOUNDARY
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Characterization of the critical Sobolev space on the optimal singularity at the origin
Get PDFIn the present paper, we investigate the optimal singularity at the origin for the functions belonging to the critical Sobolev space H^[n/p, p] (ℝ^[n]), 1 < p < ∞. With this purpose, we shall show the weighted Gagliardo-Nirenberg type inequality: ||u|| _[Lq(ℝ^[n] ; dx/|x|s)] ≤ C(1/[n-s])^[1/q + 1/p'] q^[1/p'] ||u|| ^[(n-s)p/nq] _[Lp(ℝ^[n]] ||(-Δ)^[n/2p] u||^[1-[(n-s)p]/nq] _[Lp(ℝ^[n]], (GN) where C depends only on n and p. Here, 0 ≤ s < n and p~ ≤ q < ∞ with some p~ ∈ (p, ∞) determined only by n and p. Additionally, in the case n ≥ 2 and n/[n-1] ≤ p < ∞, we can prove the growth orders for s as s ↑ n and for q as q → ∞ are both optimal. (GN) allows us to prove the Trudinger type estimate with the homogeneous weight. Furthermore, it is obvious that (GN) can not hold with the weight |x|^[n] itself. However, with a help of the logarithmic weight of the type (log 1/|x|)^[r] |x|^[n] at the origin, we cover this critical weight. Simultaneously, we shall give the minimal exponent r = [q+p']/p' so that the continuous embedding can hold
The upper bound of the best constant of Trudinger-Moser inequality and its application to Gagliardo-Nirenberg inequality
No full textOptimal embeddings of critical Sobolev-Lorentz-Zygmund spaces
Get PDFWe establish the embedding of the critical Sobolev-Lorentz-Zygmund space (Formula presented.)(ℝn ) into the generalized Morrey space M φr (ℝn ) with an optimal Young function φ. As an application, we obtain the almost Lipschitz continuity for func- tions in (Formula presented.)(ℝn ). O'Neil's inequality and its reverse play an essential role in the proofs of the main theorems.journal articl
On the optimal singularity of the critical Sobolev space and a related Sobolev type inequality with a logarithmic weight
No full textOPTIMAL EMBEDDINGS OF CRITICAL SOBOLEV-LORENTZ-ZYGMUND SPACES
No full textIn this paper, we establish the embedding on the critical Sobolev-Lorentz-Zygmund space H n p p;q;1; ;m(R n) into the generalized Morrey spaceM;r(Rn) with an optimal Young func-tion Φ. Furthermore, as an application of this embedding, we obtain the almost Lipschitz continuity for functions in H n p+1 p;q;1; ;m(R n). O’Neil’s inequality and its reverse play an essential role for the proof of main theorems
On a maximizing problem of the Sobolev embedding related to the space of bounded variation (The deepening of function spaces and its environment)
Get PDFOn the effect of equivalent constraints on a maximizing problem associated with the Sobolev type embeddings in RN
No full textIn this paper, we consider the attainability of a maximizing problem (Formula presented.)where (Formula presented.), (Formula presented.), (Formula presented.), (Formula presented.) and (Formula presented.). The existence of a maximizer for D is closely related to the exponent (Formula presented.). In fact, we show that the value (Formula presented.)is a threshold in terms of the attainability of D. © 2015 Springer-Verlag Berlin Heidelberg発行後1年より全文公
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