Sentence examples for maximal functions are from inspiring English sources

The Young functions used in this paper are Φ ( t ) = t ( 1 + log t ) r and Φ ˜ ( t ) = exp ( t 1 / r ), the corresponding average and maximal functions are denoted by ∥ ⋅ ∥ L ( log L ) r, Q, M L ( log L ) r and ∥ ⋅ ∥ exp L 1 / r, Q, M exp L 1 / r.

The Young functions to be used in this paper are Φ ( t ) = t ( 1 + log t ) r and Φ ˜ ( t ) = exp ( t 1 / r ), the corresponding average and maximal functions are denoted by ∥ ⋅ ∥ L ( log L ) r, Q, M L ( log L ) r and ∥ ⋅ ∥ exp L 1 / r, Q, M exp L 1 / r.

The Young functions to be used in this paper are Φ ( t ) = exp ( t r ) − 1 and Ψ ( t ) = t log r ( t + e ), the corresponding Φ-average and maximal functions are denoted by ∥ ⋅ ∥ exp L r, Q, M exp L r and ∥ ⋅ ∥ L ( log L ) r, Q, M L ( log L ) r.

After a careful check of its proof, we find that the condition h ∈ Δ2 is not sufficient for p ≠ 2 since the two partial maximal functions are taken supremum both j and k, it seems that if h ∈ Δ2, the partial maximal function is not pointwise controlled by the one-parameter maximal function case (line 10-13, [15]).

Weak type estimates for strong maximal functions were first studied by Jessen, Marcinkiewcz and Zygmund [11] who first proved the strong differentiation theorem.

As the selection of a given threshold is arbitrary, analyzing multiple thresholds of maximal functioning is recommended5.

Let S be a hypersurface in Rn, n ≥ 2, and dμ = ψ dσ, where ψ ∈ C∞0 (Rn) and σ denotes the surface area measure on S. Define the maximal function M associated to S and μ by [formula] It was shown by Stein that when S is the sphere in Rn, n ≥ 3, M (the spherical maximal function) is bounded on Lp(Rn) if and only if p > n/(n − 1).

(3) The Poisson maximal function is defined by M_{P}(f) (z)=sup_{t>0}biglvert P_{t}times f z bigrvert.

The maximal function is a classical tool in harmonic analysis but recently it has been successfully used in studying Sobolev functions and partial differential equations.

And the sharp maximal function is defined by M # ( f ) ( x ) = sup Q ∋ x inf c ∈ R 1 | Q | ∫ Q | f ( y ) − c | d y.

For each (P= r,Theta)inmathbf{R}^{n}-{O}), the maximal function is defined by M(P lambda,beta)=sup_{ 0< rho< frac{r}{2}}frac{lambda(B(P,rho))}{rho^{beta}}.