## On sharp higher order Sobolev embeddings.(English)Zbl 1108.46029

It is well-known that the classical Sobolev embedding theorem
$W_0^{k,p}(\Omega) \subset L^q(\Omega),\;{1\over q} = {1\over p} - {k \over n},\;1< p < {n \over k}, \tag{1}$
fails in the limit case $$p = {n \over k}$$. The main result of the present paper is an extension of (1) to this case, by replacing $$L^\infty(\Omega)$$ with a more exotic space denoted by $$L(\infty, p)(\Omega)$$. If $$1 < p \leq \infty$$, $$0 < q \leq \infty$$, $$L(p,q)(\Omega)$$ stands for the class of measurable functions such that $\int_0^\infty \{[(f^{**}(t) - f^*(t)) t^{1/p}]^q\, {dt/t}\}^{1/q} < \infty.$ Here, $$f^*$$ is the usual decreasing rearrangement function, while, for $$t > 0$$, $f^{**}(t) = {1\over t} \int_0^t f^*(u)\, du.$ The result is optimal in the sense that, if $$X(\Omega)$$ is a rearrangement invariant space containing $$W_0^{{n \over k},p}(\Omega)$$, then $$X(\Omega)$$ is contained in $$L(\infty,p)(\Omega)$$.

### MSC:

 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 26D10 Inequalities involving derivatives and differential and integral operators
Full Text:

### References:

  DOI: 10.2307/1971445 · Zbl 0672.31008  DOI: 10.1016/0362-546X(89)90043-6 · Zbl 0678.49003  DOI: 10.2307/2006999 · Zbl 0465.42015  Bennett C., Interpolation of Operators (1988) · Zbl 0647.46057  Brézis H., Comm. Partial Differential Equations pp 773–  DOI: 10.1215/S0012-7094-00-10531-5 · Zbl 1017.46023  DOI: 10.1007/BF02384772 · Zbl 1035.46502  DOI: 10.1007/BF02498218 · Zbl 0930.46027  DOI: 10.1006/jfan.1999.3508 · Zbl 0955.46019  DOI: 10.1007/BF01774283 · Zbl 0639.46034  Hansson K., Math. Scand. 45 pp 77– · Zbl 0437.31009  DOI: 10.1070/SM1969v008n02ABEH001114 · Zbl 0193.09303  DOI: 10.1070/SM1970v011n03ABEH001297 · Zbl 0216.15704  DOI: 10.1070/RM1989v044n05ABEH002287 · Zbl 0715.41050  DOI: 10.1090/S0002-9939-01-06060-9 · Zbl 0990.46022  Maz’ya V. G., Sobolev Spaces (1985)  Maz’ya V. G., Problems in Mathematical Analysis, Leningrad: LGU 3 pp 33– · Zbl 1187.42011  DOI: 10.1215/S0012-7094-63-03015-1 · Zbl 0178.47701  DOI: 10.1007/BF01305216 · Zbl 0686.46029  DOI: 10.1006/jath.2002.3730 · Zbl 1032.46045  Stein E. M., Singular Integrals and Differentiability of Functions (1970) · Zbl 0207.13501  DOI: 10.1512/iumj.1972.21.21066 · Zbl 0241.46028  G. Talenti, Nonlinear Analysis, Function Spaces and Applications 5 (Prometheus, Prague, 1995) pp. 177–230.  Trudinger N. S., J. Math. Mech. 17 pp 473–  Yudovich V., Dokl. Akad. Nauk SSSR 138 pp 805–
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.