H-Distributions: An Extension of H-Measures to an L-L Setting

and Applied Analysis 3 can be extended to a continuous mapping Tφ : L d → L d . One calls operator Tφ the L-multiplier operatorwith symbol φ. Theorem 1.4 Hörmander-Mikhlin . Let φ ∈ L∞ d have partial derivatives of order less than or equal to κ, where κ is the least integer strictly greater than d/2 (i.e., κ d/2 1). If, for some constant k > 0, ∀r > 0 ( ∀α ∈ 0 ) |α| ≤ κ ⇒ ∫ r/2≤|ξ|≤r ∣∣ ∣D ξφ ξ ∣∣ ∣ 2 dξ ≤ k2rd−2|α|, 1.6 then, for any p ∈ 〈1,∞〉 and the associated multiplier operator Tφ, there exists a constant Cd (depending only on the dimension d; see [18, page 367]) such that ∥Tφ ∥∥ Lp →Lp ≤ Cd max { p, 1 p − 1 } ( k ∥φ ∥∥ ∞ ) . 1.7 Remark 1.5. It is important to notice that, according to 19, Section 3.2, Example 2 , if the symbol of a multiplier is aC function defined on the unit sphere Sd−1 ⊆ d , then the constant k from Theorem 1.4 can be taken to be equal to ‖φ‖Cκ Sd−1 . By an application of Theorem 1.4, in Section 2 we are able to introduceH-distributions see Theorem 2.1 —an extension of H-measures in the L-setting. Its proof is the main result of the paper and forms Section 3. We conclude in Section 4 by an L-variant of the localisation principle and a proof of an L, L ′ -variant of the div-curl lemma. Remark 1.6. Recently, variants of H-measures with a different scaling were introduced the parabolic H-measures 15, 20 and the ultraparabolic H-measures 21 . We can apply the procedure from this paper to extend the notion of such H-measures to the L-setting in the same fashion as it is given here based on Theorem 1.1 for the classicalH-measures. 2. A Generalisation of H-Measures We have already seen Remark 1.2 that an H-measure μ corresponding to a sequence un in L2 d can describe its loss of strong compactness. We would like to introduce a similar notion describing the loss at least in Lloc of strong compactness for sequences weakly converging in L d . Consider a sequence un weakly converging to zero in L d and satisfying the following sequence of differential equations: