A trace problem for associate Morrey potentials

1.1 Description of Question 1.1

Stemming from an extensive investigation of [2, 7, 9, 10, 45], this paper attempts to answer the coming-up-next restriction question, i.e., a trace problem for associate Morrey potentials. Namely, the Riesz images Iα⁢H1p,κ⊂Iα⁢Hp,κ of the spaces H1p,κ⊂Hp,κ associated with the Morrey spaces L∞pp-1,κ⊃Lpp-1,κ that have been proven to be of independent interest in geometric potential analysis (see, e.g., [5]) but also of great value in treating the local behaviour of an appropriate solution to certain elliptic partial differential equation or the incompressible Navier-Stokes system (see, e.g., [33, 29, 21, 22, 42, 43, 16, 34, 39, 38, 32, 28, 31, 20, 35, 11, 14, 15]):

A resolution of the above question relies on the following fundamentals.

First of all, from a geometric measure theory perspective, a Radon measure on ℝn is defined as a Borel regular measure assigning finite mass to any compact subset of ℝn (cf. [18, p. 5]). For a given Radon measure μ on ℝn and a nonnegative number β, denote by

the β-th order ball variation of μ. Trivially, |||μ|||0 is the total variation of μ. Obviously, this variation |||μ|||β is finite whenever β∈{1,…,n} and μ is the β-dimensional Hausdorff measure ℋβ restricted on a smooth manifold 𝕄⊂ℝn with ℋβ⁢(𝕄)<∞.

Next, given α∈(0,n) the Riesz operator Iα acting on a Lebesgue ν-measurable function f in ℝn is determined by

whose constant multiple

defines an inverse for a power of the Laplace operator on ℝn (cf. [5, p. 9]), i.e., such an singular integral u solves the α2-th order Laplace equation (-Δ)α2⁢u=f using the Fourier transform.

After that, for (q,λ)∈[1,∞)×(0,n], let Lμ,∞q,λ be the weak Morrey space of all μ-measurable functions f on ℝn with

where B⁢(x,r) is the Euclidean open ball with centre x and radius r. Naturally, as a proper subspace of Lμ,∞q,λ, the symbol Lμq,λ is employed to represent the Morrey space of all μ-measurable functions f with

In particular, Lμ,∞q=Lμ,∞q,n and Lμq=Lμq,n. Moreover, if μ=ν (the n-dimensional Lebesgue measure on ℝn) and (q,λ)=(p,κ)∈[1,∞)×(0,n], then Lμ,∞q,λ and Lμq,λ will be simply written as L∞p,κ and Lp,κ respectively, and the endpoint spaces L∞p,n and Lp,n coincide with the weak-Lp space L∞p and Lp, respectively; but if q=∞, then L∞∞,κ=L∞,κ=L∞. Furthermore, for p∈[1,∞] and p′≡p/(p-1) with ∞≡1/(1-1) and 1≡∞/(∞-1), set

Naturally, H1p,κ and Hp,κ are called the associate spaces [L∞p′,κ]⋆ and [Lp′,κ]⋆ of the Morrey spaces L∞p′,κ and Lp′,κ, respectively (cf. [36]). Especially, one has H11,κ=H1,κ=L1 (for 0<κ≤n) and H1∞,n⊊H∞,n=L∞ (due to L1=L1,n⊊L∞1,n, cf. [17]).

Last but not least, note that x↦|x|-κp is an element of Lp,κ for 0≤κ<n (cf. [46, p. 587]) and that x↦(1+|x|η)-1 belongs to Hp,κ for η>κp (cf. [24, Example 2.5]). So, the next implication seems to be very natural and motivative (cf. [26, p.85] and [6, Theorem 3.3]):

In the above and below, X≳Y stands for that there exists a constant c>0 such that X≥c⁢Y, X≳Y≳X will be simply written as X≈Y, and 1E represents the characteristic function of a set E⊆ℝn.

1.2 Answering Question 1.1

Working on the above question plus the last implication one discovers the following trace and embedding principle for the associate Morrey potentials Iα⁢Hp,κ (regarded as the space of tempered distributions having derivatives of order α in the associate Morrey space Hp,κ).

Interestingly, as an immediate application of Theorem 1.2, the next corollary completely describes the embedding of the k-th order associate Morrey–Sobolev space into Lμq,λ. More precisely, if

then

for all C∞-functions f with compact support in ℝn (see, e.g., [4, p. 393] or [30, p. 19]), and hence applying Theorem 1.2, together with a small modification of the argument used in the proof of [30, p. 71, Theorem] as p=1, produces the following corollary.

Through exploiting the local nature of some indicator-like functions, introducing Lemmas 2.1, 3.1 and 4.1, and utilising the dual structure of a local Lμq-space, the non-trivial arguments for (i), (ii) and (iii) of Theorem 1.2 and their crucial Remarks 2.2, 3.2 and 4.2 will be arranged in Sections 2, 3 and 4, respectively. It is believed that the techniques developed in the forthcoming three sections can be applied to investigate a similar trace problem for the following Cordes–Nirenberg potentials (see, e.g., [26, 27, 16, 34]):

Nevertheless, this investigation will be the subject of a future article.

2.1 A lemma for Theorem 1.2(i)

To prove Theorem 1.2 (i), one needs the following result which improves [2, Theorem 5.1].