Perron's integral for derivatives in $L^{r}$

Gordon, Leo A.

doi:10.4064/sm-28-3-295-316

Cited by 14 publications

(17 citation statements)

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“…To establish pointwise estimates for solutions of elliptic partial differential equations, in 1961 Calderon and Zygmund introduced the L r -derivative (see [3]) and in 1968 L. Gordon described a Perron-type integral, the P rintegral, that recovers a function from its L r -derivative (see [4]). In 2004, Musial and Sagher extended the P r -integral to the L r -Henstock-Kurzweil integral, the HK r -integral, that recovers also a function from its L r -derivative (see [6]).…”

Section: History and Aimmentioning

confidence: 99%

The $$L^{r}$$-Variational Integral

Tulone

Musial

2022

Mediterr. J. Math.

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We define the $$L^r$$ L r -variational integral and we prove that it is equivalent to the $$HK_r$$ H K r -integral defined in 2004 by P. Musial and Y. Sagher in the Studia Mathematica paper The$$L^{r}$$ L r -Henstock–Kurzweil integral. We prove also the continuity of $$L^r$$ L r -variation function.

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Section: History and Aimmentioning

confidence: 99%

The $$L^{r}$$-Variational Integral

Tulone

Musial

2022

Mediterr. J. Math.

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“…We recall here the main definitions and facts related to the notion of the HK r -integral (see [3,6]).…”

Section: Preliminariesmentioning

confidence: 99%

“…This L r -derivative was introduced in [1] by Calderón and Zygmund to be used in some estimates for solutions of elliptic partial differential equations. The HK r -integral was defined as an extension of a Perron-type integral, the P r -integral, which was defined earlier by L. Gordon [3] and which also recovers a function from its L r -derivative. The HK r -integral turned out to be strictly wider than the P r -integral (see [7]).…”

Section: Introductionmentioning

confidence: 99%

On Descriptive Characterizations of an Integral Recovering a Function from Its $$L^r$$-Derivative

2022

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It is proved that any function of a Lusin-type class, the class of ACGr-functions, is differentiable almost everywhere in the sense of a derivative defined in the space L r , 1 ≤ r < ∞. This leads to obtaining a full descriptive characterization of a Henstock-Kurzweil-type integral, the HKrintegral, which serves to recover functions from their L rderivatives. The class ACGr is compared with the classical Lusin class ACG and it is shown that a continuous ACGfunction can fail to be L r -differentiable almost everywhere.

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“…Here again, if we wish to generalize Theorem 1 we need/and <p that change sign often, for S. Foglio [17] showed that if /s* 0 and g is monotone increasing, the A^-Perron integral of f(t) with respect to g(t) in [22] is also a Lebesgue-Stieltjes integral. There are other generalized Perron-type integrals, such as integrals between Perron's and the £esaro-Perron, defined by L. Gordon [19] using for his upper right derivate the greatest lower bound of all constants a for which…”

Section: ) Lim (5) Fn + (« V T)(f(t) -F(u)) Dt/ (S) Fn + (U V T)mentioning

confidence: 99%

“…q(t + h)-q{t) > {P(t + h)~ P(t) -f(t)h) (a<t<t + h^b,O<h< 8 2 ( 0 ) . Replacing 8,, 8 2 by 8 = min(8,, 8 2 ), as Q, q are monotone increasing,(18,19) give…”

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A problem in two-dimensional integration

Henstock

1983

J Aust Math Soc A

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If two functions of a real variable are integrable over two intervals, say of t, T, respectively, then the product of the two functions should be integrable over the rectangular product of the two intervals of t and T. For the Lebesgue integral, definable using non-negative functions alone, the proof is easy. For non-absolute integrals such as the Perron, Cesaro-Perron, and Marcinkiewicz-Zygmund integrals we have difficulties since the functions cannot be assumed non-negative. But the present paper gives a proof. [a, b], [a, /?] of the real line, respectively, then f(t) that is, (1) / f(t) 0.[26], page 63, Theorem 7, is more general, but the assumption of bounded variation shows that we are again dealing with absolute integrals. The problem ought to be easy even when the integrals are not absolute, but the changing of sign of / and

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Perron's integral for derivatives in $L^{r}$

Cited by 14 publications

References 0 publications

The $$L^{r}$$-Variational Integral

The $$L^{r}$$-Variational Integral

On Descriptive Characterizations of an Integral Recovering a Function from Its $$L^r$$-Derivative

A problem in two-dimensional integration

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