Bounds on layer potentials with rough inputs for higher order elliptic equations

Abstract

In this paper, we establish square‐function estimates on the double and single layer potentials with rough inputs for divergence form elliptic operators, of arbitrary even order 2m, with variable t‐independent coefficients in the upper half‐space.

Nontangential Estimates on Layer Potentials and the Neumann Problem for Higher-Order Elliptic Equations

“…See [24]. We remark that this definition coincides with the definition of S Lġ involving the Newton potential given in [19,20]. In [23] we showed that S L extends by density to an operator defined on all of L p (R n ) for p sufficiently close to 2 and that satisfies the bound (1.19).…”

“…See, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. t-independent coefficients in the higher order case have received much more limited study; Hofmann and Mayboroda together with the author of the present paper have begun their study in [19][20][21][22][23]. Specifically, in [22], we established the following result.…”

“…We now recall the operator S L ∇ of [20,23]. If |β| = m and β n+1 ≥ 1, and if h ∈ L 2 (R n ), then we let…”

“…We remark that if A is t-independent, then N + (∇ m−1 v) ∈ L 2 (R n ) is a stronger statement than sup t>0 ∇ m−1 v( · , t) L 2 (R n ) < ∞; see [20,Lemma 3.20], reproduced as Lemma 3.2 below. The first of the two main results of the present paper is the following theorem.…”

“…The main results of [19,20] were the p ≤ 2 cases of the following estimates on layer potentials. (The p > 2 cases were established later in [23].…”