Some Remarks Concerning Potentials on Different Spaces

Here dz denotes Lebesgue measure on R. More precisely, if f lies in L(R), then P (f) is defined almost everywhere on R if 1 ≤ q < n, it is defined almost everywhere modulo constants when q = n, and it is defined modulo constants everywhere if n < q < ∞. (If q = ∞, then one can take it to be defined modulo affine functions.) We shall review the reasons behind these statements in a moment. The case where n = 1 is a bit different and special, and we shall not pay attention to it in these notes for simplicity. Similarly, we shall normally restrict our attention to functions in L with 1 < q < ∞. A basic fact about this operator on R is that if f ∈ L(R), then the first derivatives of P (f), taken in the sense of distributions, all lie in L(R), as long as 1 < q < ∞. Indeed, the first derivatives of P (f) are given by first Riesz transforms of f (modulo normalizing constant factors), and these are well-known to be bounded on L when 1 < q < ∞. (In connection with these statements, see [27, 28].) One might rephrase this as saying that P maps L into the Sobolev space of functions on R whose first derivatives lie in L when 1 < q < ∞. Instead of taking derivatives, one can look at the oscillations of P (f) more directly, as follows. Let r be a positive real number, which represents the scale at