Generalization of Frostman's Lemma?

Suppose that K is a compact subset of R^d , a > 0. Frostman's Lemma
says that the Hausdorff measure m_a(K) is positive if and only if
there
exists a probability measure mu supported on K such that

(*) mu(B(x,r)) < c r^a

for every x in R^d and r > 0.

What if we consider instead an L^p condition:

(**) {int_R^d mu(B(x,r))^p dx}^{1/p} <= c r^a

? Do we know that K satisfies some covering condition if and
only if there exists a mu satisfying (**)?

Dato

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