The Caffarelli-Kohn-Nirenberg type inequalities involving critical and supercritical weights

On radial symmetry and its breaking in the Caffarelli-Kohn-Nirenberg type inequalities for <i>p </i>= 1

“…The assertion 1 of this theorem is already established in Proposition 3.1; [21]. The assertions 2 and 3 are proved in section 3 by virtue of Theorem 1.2.…”

“…These estimates are simple variants of the assertions 4 and 5 of Theorem 2.2 in [21]. For reader's convenience, we give a simple proof of the assertion 1 in Section 2 as Proposition 2.2 (see also Proposition 3.2; [21]).…”

“…As was mentioned before, when γ > 0 holds, the CKN-type inequalities were introduced in [4] as a part of multiplicative interpolation inequalities, and later in [21] the CKN-type inequalities were further investigated for all γ ∈ R. In the present paper, we shall give a simple proof of the CKN-type inequalities (1.2) with p = 1 and the best constant S 1,q ;γ defined below, using equivalence between the CKN-type inequalities with p = 1 and isoperimetric inequalities with weights. We note that in [18] a class of weighted Sobolev inequalities were already studied by using isoperimetric inequalities with general weight functions.…”

“…[10,12,13,14,15] and see also [3,20] for p > 1). Recently in [21] the second author of the present paper systematically investigated the CKN-type inequalities involving critical and supercritical weights. More precisely, validity of inequalities, existence of extremal functions, continuity of best constants, symmetry of extremal functions and symmetry breaking of the best constants were established.…”

“…More precisely, validity of inequalities, existence of extremal functions, continuity of best constants, symmetry of extremal functions and symmetry breaking of the best constants were established. For this purpose the condition p > 1 was assumed in [21] in most cases, but some results on symmetry were studied including the case where p = 1 as well. On a basis of these observation, we study in the present paper symmetry property of the best constant and its breaking phenomena for the CKN-type inequalities when p = 1.…”